The Catholic Church is by far the oldest and largest branch of Christianity that does not adhere to the doctrine of sola scriptura.
The doctrine of sola scriptura proclaims the canonical Scriptures, especially the New Testament, to be the only infallible source and rule of faith and practice, and asserts the right of private interpretation of the same, in distinction from the Roman Catholic view, which declares the Bible and tradition to be co-ordinate sources and rule of faith, and makes tradition, especially the decrees of popes and councils, the only legitimate and infallible interpreter of the Bible. In its extreme form Chillingworth expressed this principle of the Reformation in the well-known formula, "The Bible, the whole Bible, and nothing but the Bible, is the religion of Protestants." Protestantism, however, by no means despises or rejects church authority as such, but only subordinates it to, and measures its value by, the Bible, and believes in a progressive interpretation of the Bible through the expanding and deepening consciousness of Christendom.
The Catholic Church has traditionally taught that there are three authorities in a Christian's life:
1 - Sacred Scripture
2 - Sacred Tradition
3 - Magisterium (teaching office)
These 3 authorities are like a 3 legged stool holding up and preserving the teachings of Christ.
Sacred Tradition (with a capital "T") refers to the doctrines that have been preserved and handed down "whether by word of mouth or by letter" (2 Thess. 2:15). Examples of these universally accepted Traditions that are not explicitly defined in Sacred Scripture include the doctrines of the Trinity and the Incarnation. This does not refer to traditions with a lower case "t" (i.e. Christmas date, liturgical songs, etc.). This distinction should differentiate between Patristic Writings like those of St. Augustine, and an excerpt from G. K. Chesterton's "Orthodoxy."
The Catechism of the Catholic Church states in Part I - Section I clearly states:
THE RELATIONSHIP BETWEEN TRADITION AND SACRED SCRIPTURE
One common source. . .
80 "Sacred Tradition and Sacred Scripture, then, are bound closely
together, and communicate one with the other. For both of them,
flowing out from the same divine well-spring, come together in some
fashion to form one thing, and move towards the same goal." Each of
them makes present and fruitful in the Church the mystery of Christ,
who promised to remain with his own "always, to the close of the age".
. . . two distinct modes of transmission
81 "Sacred Scripture is the speech of God as it is put down in writing
under the breath of the Holy Spirit."
"And [Holy] Tradition transmits in its entirety the Word of God which
has been entrusted to the apostles by Christ the Lord and the Holy
Spirit. It transmits it to the successors of the apostles so that,
enlightened by the Spirit of truth, they may faithfully preserve,
expound and spread it abroad by their preaching."
82 As a result the Church, to whom the transmission and interpretation
of Revelation is entrusted, "does not derive her certainty about all
revealed truths from the holy Scriptures alone. Both Scripture and
Tradition must be accepted and honored with equal sentiments of
devotion and reverence."
Apostolic Tradition and ecclesial traditions
83 The Tradition here in question comes from the apostles and hands on
what they received from Jesus' teaching and example and what they
learned from the Holy Spirit. The first generation of Christians did
not yet have a written New Testament, and the New Testament itself
demonstrates the process of living Tradition.
Tradition is to be distinguished from the various theological,
disciplinary, liturgical or devotional traditions, born in the local
churches over time. These are the particular forms, adapted to
different places and times, in which the great Tradition is expressed.
In the light of Tradition, these traditions can be retained, modified
or even abandoned under the guidance of the Church's Magisterium.
The Magisterium plays the part of "Authoritative Interpreter" or "Home Plate Umpire," by making the final decision on divisive doctrinal matters (i.e. Luther and Calvin's veracious dispute about the substantive reality of the Eucharist).
The Church has claimed from the first century that all 3 of these authoritative faculties have been ultimately guided by the Holy Spirit, or "Paraclete" promised by Christ. There are hundreds of documents written by Early Church Fathers emphasizing the necessary role of Sacred Tradition within the context of Divine Revelation.
St. Augustine writes:
“But in regard to those observances which we carefully attend and
which the whole world keeps, and which derive not from Scripture but
from Tradition, we are given to understand that they are recommended
and ordained to be kept, either by the apostles themselves or by
plenary [ecumenical] councils, the authority of which is quite vital
in the Church” (Letter to Januarius [A.D. 400]).
For my part, I should not believe the gospel except moved by the
authority of the Catholic Church. So when those on whose authority I
have consented to believe in the gospel tell me not to believe in
Manicheus, how can I but consent?" C. Epis Mani 5,6
Martin Luther echoes Augustine in one of commentaries on Sacred Scripture:
“Yes, we ourselves find it difficult to refute it, especially since we
concede—as we must—that so much of what they say is true: that the
papacy has God’s Word and the office of the apostles, and that we have
received Holy Scripture, Baptism, the Sacrament, and the pulpit from
them. What would we know of these if it were not for them? [LW 24:304]
Sola scriptura may be contrasted with prima scriptura, which holds that, besides canonical scripture, there are other guides for what a believer should believe, and how he or she should live. Examples of this include the general revelation in creation, traditions, charismatic gifts, mystical insight, angelic visitations, conscience, common sense, the views of experts, the spirit of the times or something else. Prima scriptura suggests that ways of knowing or understanding God and his will, that do not originate from canonized scripture, are in a second place, perhaps helpful in interpreting that scripture, but testable by the canon and correctable by it, if they seem to contradict the scriptures.
This concludes the first half of this answer. It has presented the perspective of the Catholic Church's stance on the role of extra biblical texts in defining beliefs.
The second half of this answer is intended to present Biblical "support" for the doctrine of prima scriptura.
Below are quotes from Sacred Scripture that support prima scriptura. These exerpts are not intended to be used as proofs, but support.
Gen. to Rev. - Scripture never says that Scripture is the sole
infallible authority for God's Word. Scripture also mandates the use
of tradition. This fact alone disproves sola Scriptura.
Matt. 28:19; Mark 16:15 - those that preached the Gospel to all
creation but did not write the Gospel were not less obedient to Jesus,
or their teachings less important.
Matt. 28:20 - "observe ALL I have commanded," but, as we see in John
20:30; 21:25, not ALL Jesus taught is in Scripture. So there must be
things outside of Scripture that we must observe. This disproves
"Bible alone" theology.
Mark 16:15 - Jesus commands the apostles to "preach," not write, and
only three apostles wrote. The others who did not write were not less
faithful to Jesus, because Jesus gave them no directive to write.
There is no evidence in the Bible or elsewhere that Jesus intended the
Bible to be sole authority of the Christian faith.
Luke 1:1-4 - Luke acknowledges that the faithful have already received
the teachings of Christ, and is writing his Gospel only so that they
"realize the certainty of the teachings you have received." Luke
writes to verify the oral tradition they already received.
John 20:30; 21:25 - Jesus did many other things not written in the
Scriptures. These have been preserved through the oral apostolic
tradition and they are equally a part of the Deposit of Faith.
Acts 8:30-31; Heb. 5:12 - these verses show that we need help in
interpreting the Scriptures. We cannot interpret them infallibly on
our own. We need divinely appointed leadership within the Church to
Acts 15:1-14 – Peter resolves the Church’s first doctrinal issue
regarding circumcision without referring to Scriptures.
Acts 17:28 – Paul quotes the writings of the pagan poets when he
taught at the Aeropagus. Thus, Paul appeals to sources outside of
Scripture to teach about God.
1 Cor. 5:9-11 - this verse shows that a prior letter written to
Corinth is equally authoritative but not part of the New Testament
canon. Paul is again appealing to a source outside of Scripture to
teach the Corinthians. This disproves Scripture alone.
1 Cor. 11:2 - Paul commends the faithful to obey apostolic tradition,
and not Scripture alone.
Phil. 4:9 - Paul says that what you have learned and received and
heard and seen in me, do. There is nothing ever about obeying
Col. 4:16 - this verse shows that a prior letter written to Laodicea
is equally authoritative but not part of the New Testament canon. Paul
once again appeals to a source outside of the Bible to teach about the
Word of God.
1 Thess. 2:13 – Paul says, “when you received the word of God, which
you heard from us..” How can the Bible be teaching first century
Christians that only the Bible is their infallible source of teaching
if, at the same time, oral revelation was being given to them as well?
Protestants can’t claim that there is one authority (Bible) while
allowing two sources of authority (Bible and oral revelation).
1 Thess. 3:10 - Paul wants to see the Thessalonians face to face and
supply what is lacking. His letter is not enough.
2 Thess. 2:14 - Paul says that God has called us "through our Gospel."
What is the fullness of the Gospel?
2 Thess. 2:15 - the fullness of the Gospel is the apostolic tradition
which includes either teaching by word of mouth or by letter.
Scripture does not say "letter alone." The Catholic Church has the
fullness of the Christian faith through its rich traditions of
Scripture, oral tradition and teaching authority (or Magisterium).
2 Thess 3:6 - Paul instructs us to obey apostolic tradition. There is
no instruction in the Scriptures about obeying the Bible alone (the
word "Bible" is not even in the Bible).
1 Tim. 3:14-15 - Paul prefers to speak and not write, and is writing
only in the event that he is delayed and cannot be with Timothy.
2 Tim. 2:2 - Paul says apostolic tradition is passed on to future
generations, but he says nothing about all apostolic traditions being
eventually committed to the Bible.
2 Tim. 3:14 - continue in what you have learned and believed knowing
from whom you learned it. Again, this refers to tradition which is
found outside of the Bible.
James 4:5 - James even appeals to Scripture outside of the Old
Testament canon ("He yearns jealously over the spirit which He has
2 Peter 1:20 - interpreting Scripture is not a matter of one's own
private interpretation. Therefore, it must be a matter of "public"
interpretation of the Church. The Divine Word needs a Divine
Interpreter. Private judgment leads to divisions, and this is why
there are 30,000 different Protestant denominations.
2 Peter 3:15-16 - Peter says Paul's letters are inspired, but not all
his letters are in the New Testament canon. See, for example, 1 Cor.
5:9-10; Col. 4:16. Also, Peter's use of the word "ignorant" means
unschooled, which presupposes the requirement of oral apostolic
instruction that comes from the Church.
2 Peter 3:16 - the Scriptures are difficult to understand and can be
distorted by the ignorant to their destruction. God did not guarantee
the Holy Spirit would lead each of us to infallibly interpret the
Scriptures. But this is what Protestants must argue in order to
support their doctrine of sola Scriptura. History and countless
divisions in Protestantism disprove it.
1 John 4:1 - again, God instructs us to test all things, test all
spirits. Notwithstanding what many Protestants argue, God's Word is
not always obvious.
1 Sam. 3:1-9 - for example, the Lord speaks to Samuel, but Samuel
doesn't recognize it is God. The Word of God is not self-attesting.
1 Kings 13:1-32 - in this story, we see that a man can't discern
between God's word (the commandment "don't eat") and a prophet's
erroneous word (that God had rescinded his commandment "don't eat").
The words of the Bible, in spite of what many Protestants must argue,
are not always clear and understandable. This is why there are 30,000
different Protestant churches and one Holy Catholic Church.
Gen. to Rev. - Protestants must admit that knowing what books belong
in the Bible is necessary for our salvation. However, because the
Bible has no "inspired contents page," you must look outside the Bible
to see how its books were selected. This destroys the sola Scriptura
theory. The canon of Scripture is a Revelation from God which is
necessary for our salvation, and which comes from outside the Bible.
Instead, this Revelation was given by God to the Catholic Church, the pinnacle and foundation of the truth (1 Tim. 3:15).