Sola Scriptura isn't taught in the Bible
Sola scriptura is a self-refuting doctrine. Rather, we find the following biblical instructions to maintain traditions that were not passed down in biblical writings:
"I commend you because you remember me in everything and maintain the traditions just as I handed them on to you" (1 Corinthians 11:2, NRSV).
"So then, brothers and sisters, stand firm and hold fast to the traditions that you were taught by us, either by word of mouth or by our letter" (2 Thessalonians 2:15, NRSV).
In other words, the Bible itself teaches Scripture plus tradition(s).
Some might argue that verses such as 2 Timothy 3:16-17 or Revelation 22:18-19 provide support for sola scriptura, but it should be kept in mind that γραφή (graphē | "writings"/"Scripture") almost always refers to the Hebrew Bible ("Old Testament").1
Sola scriptura did not originate until the 14th century and did not become widespread until the 16th century. It's already been established that the Bible does not teach sola scriptura, so where did it come from? It was clearly invented by men (Wycliffe, Luther, et. al.).
Most Christians didn't have Bibles
The books of the New Testament were likely not completed until the end of the first century.2 This means there were approximately 60 years between Christ's ascension and the completion of all of the works of the New Testament.
But even then, those works weren't compiled into any official canon until hundreds of years later in history, and many Christians only had some of the works and often several that are not part of the modern Christian canon. Also, the concept of 'canon' at that time had little to do with determining what books had 'authority' or were 'Scriptural' (that word simply means 'writings'), but rather which books would be read during public worship (in fact, Revelation was always excluded from this list until much later in history). It should also be noted that the 'Hebrew Bible' read by most Christians for hundreds of years was the Greek Septuagint which contained numerous books that are not in the modern Protestant biblical canon, and the text conflicts with the Hebrew/Masoretic in numerous places (and this is also the version of the Bible most frequently cited in the New Testament—not the Hebrew version used by Protestants today). In fact, Jesus himself quoted books that have been excluded from Protestant Bibles.3
To complicate things even further, literacy rates were generally low until the medieval period in the West (and history shows that heresy and higher literacy rates are strongly correlated4). Add to this the fact that books were generally too expensive for the common person until after the invention of the printing press in the 15th century, so for most of Church history, the majority of Christians have not had access to a Bible (including many clergy). This means that Christianity was largely passed on through tradition and word-of-mouth instruction for most of its history.
The Church produced the Bible—not vice versa
As evidenced above, the Church predates the formation of the Bible. Not only were there numerous books that were disputed and read in many Christian communities (even to this day), but there were also notable manuscript differences between existing copies of texts and vast differences between various translations of those texts and their supposed originals. In addition, the formation of the biblical canon was a long and complicated process with underlying pragmatic, theological, and even political concerns. Ultimately, various factions of the Christian church decided which books were 'canonical'—and there is still disagreement among Christians on which texts these are to this day.5 The basis for these decisions was usually whether or not these texts were faithful to the apostolic tradition(s) that has been passed down in the Church (keep in mind that no one agreed on which texts were 'authoritative' yet).
Proponents of sola scriptura must trust that the Church got the biblical canon right. If the books were 'self-authenticating' (as many Protestants like to claim), then why would so many Christians have disputed their status, to include folks like Athanasius, Jerome, and Augustine? Why would 'self-authenticating' works take hundreds of years to be accepted? The 'self-authentication' idea simply does not hold up under historical scrutiny. The Bible itself contains no authoritative list of what books are 'Scriptural' / 'canonical' or not (and even if it did, claims of authenticity are not guarantees of it, or else numerous Gnostic texts that the Church unanimously excluded would not have been).
None of the original biblical manuscripts are extant
All that we have are copies of copies of.... We have to rely on the copyists. The field of textual criticism has shown that copyists were far from being inerrant and infallible. The manuscript evidence shows that scribes sometimes modified the biblical texts to harmonize passages, to correct historical errors, and to establish later theological doctrines. The Bible itself makes no claims about God divinely protecting the transmission of manuscripts, and even if it did, it is clear that there are numerous transmission errors. Granted, many of these differences do not affect major teachings or sections of text, but some do. Even if one insists on God's preservation of the biblical texts, then would he not also perfectly preserve Jesus and the apostles' oral teachings as well?
The reality is that many proponents of sola scriptura also believe in an inerrant and infallible Bible that doesn't exist. It's just as much a leap of faith as belief in the preservation of apostolic tradition(s).
The 'fruit' of Sola Scriptura is division and disunity
Jesus taught that one of the criteria for identifying false prophets is to watch for 'bad fruit' (Matthew 7:15-20), and he prayed to his Father for his disciples "that they may all be one... so that the world may believe that you have sent me" (John 17:21, NRSV). The apostle Paul also often exhorted Christians to remain in peace and unity (1 Corinthians 1:10-17, Ephesians 4:1-16), pointing out that
"There is one body and one Spirit, just as you were called to the one hope of your calling, one Lord, one faith, one baptism, one God and Father of all, who is above all and through all and in all" (Ephesians 4:4-6, NRSV).
Given that there were at least 33,000 Protestant denominations in the US alone in 2005, it's safe to say that Protestantism is extremely divided. Granted, Roman Catholicism and Eastern Orthodoxy are not without their own controversies and divisions, but they handle them quite differently—generally remaining Catholic/Orthodox despite disagreement on some issues (and there is rarely disagreement on the big issues such as salvation, Eucharist, justification, etc. as in Protestantism, where most groups have distinctive teachings about such issues). This shows that Protestant teaching on many issues continuously changes over time, showing that biblical interpretation within a sola scriptura framework is often subjective and mutable. Indeed there are few Protestants who even maintain the teachings of their Reformation forebears (e.g. Luther believed in the perpetual virginity of Mary).
Sola scriptura is a logical fallacy
In addition to the self-refuting axiom that the Bible itself doesn't teach sola scriptura (and thus it is a man-made doctrine), the argument can be shown to rely on faulty assumptions as well. A fundamental assumption of proponents is that the meaning of Scripture is clear enough that anyone could understand it by simply reading it, and thus no one needs the Church's (or any authority's) help in the process (despite the text itself saying the exact opposite about itself in Acts 8:26-40 and 2 Peter 3:161). If this were true, then why are there 33,000+ Protestant denominations all claiming to correctly interpret the Bible?
The reality is that subjective human reason is the ultimate authority for proponents of sola scriptura. One cannot read a text without having first learned the language in which it was written (this alone brings a huge subjective bias/context to the text, hence the study of hermeneutics). Since a majority of Bible-readers do not know Greek, Hebrew, nor Aramaic, most of them rely on translations. Setting aside translation bias (another significant factor), this means that most readers use a secondary source to read the text. On top of this, most folks are largely ignorant of the historical, linguistic, and literary setting(s) forming the context(s) of the biblical text(s), and scholars are continuously learning new things about these contexts that affect biblical interpretation. When variant readings are present, or a variety of possible meaning is present in a text, how is the correct interpretation decided? By subjective human reason and the choice of the individual. Thus sola scriptura elevates human reason as the final authority over the meaning of the text (regardless of how thousands of years of Christians have interpreted it in the past).
1 The notable exception is 2 Peter 3:16, which is likely pseudepigraphal and most likely written much later than a majority of other New Testament texts. Even so, scholars who accept 2 Peter as 'canonical' also recognize that γραφή refers exclusively to the Hebrew Bible for most NT authors. Even if it doesn't, much of the NT was not yet written nor compiled at the time of many such references (how could Timothy know Scriptures 'from infancy' that hadn't been written until he was a young adult?). In addition, 2 Peter 3:16 actually teaches that some texts are hard to understand, and 1:20 teaches that individuals should not derive the meaning of prophecy from private interpretation. Combined with texts such as Acts 8:26-40, a strong case could even be made that the Bible teaches that Scripture requires proper interpretation—and not privately by individuals.
2 I'm aware of the numerous disputes about the dating of various texts. For a hyper-conservative estimate, subtract 30 years from my calculations.
3 For some examples, compare the following passages: Matt. 6:14-15 with Ecclesiasticus (Sirach) 28:2; Matt. 6:7 with Ecclesiasticus (Sirach) 7:15(14); Matt. 7:12 with Tobit (Tobias) 4:16(15); Luke 12:18-20 with Sirach 11:19 (Ecclus. 11:19-20); Acts 10:34 with Ecclus. 35:15 (Sirach 35:12); Acts 10:26 with Wisdom 7:1; and Matt. 8:11 with Baruch 4:37.
4 Yes, yes, I know. Correlation does not equal causation. To evaluate this claim for yourself, cf. Peter Biller & Anne Hudson. Heresy and Literacy, 1000-1530 (Cambridge Studies in Medieval Literature) (Cambridge University Press, 1996). Cf. also Brian Stock. The Implications of Literacy: Written Language and Models of Interpretation in the Eleventh and Twelfth Centuries (Princeton University Press, 1987). It is interesting that the only historical proponents of arguing solely from Scripture were heretics (and even Protestants consider most of them to be heretics).
5 Cf. Lee Martin McDonald. The Biblical Canon: Its Origin, Transmission, And Authority, 3rd ed. (Hendrickson Publishers, 2007). McDonald is an evangelical scholar. There is much more dissenting scholarship on the issue but this is a good start.