How does the Bible teach us to direct our prayers?
In Scripture we read in the Lord's Prayer that we should pray in this manner,
"Our Father which art in heaven..." (Matthew 6:9) and in John 16 that we should pray in Jesus' name. We can also mention 1st Peter 1:17 in this context,
"And if ye call on the Father, who without respect of persons judgeth according to every man's work, pass the time of your sojourning here in fear:" which assumes that prayer is offered to the Father. There is nothing else taught on this subject. However, both Stephen and John do make a short exclamatory sort of prayer to Jesus after seeing a vision of Jesus, namely in Acts 7:59
"Lord Jesus, receive my spirit." and Revelation 22:20
"Even so, come, Lord Jesus." There are no prayers addressed to the Holy Ghost in scripture that I can see (correct me in the comments if you know of one).
What is an overview of the practices of Christian denominations regarding the proper direction of prayer?
I don't think the difference in practice today breaks down by denomination so much as the different views on the Trinity.
In the church of Christ we pray only to the Father in Jesus' name (although I've heard some in public prayer throw in a "Jesus we love you"). But we aren't credal, so we don't accept or enforce any creed like the Athanasian Creed. We go by that motto "no creed but the Bible." As a result there are different views on the Trinity held among us, but in the mainstream I'd say nobody is an Athanasian Trinitarian but only Subordinationist Trinitarians. That is, rather than holding the Father and Son to be "coequal" (as in the Athanasian Creed) or that "in this Trinity none is afore or after another; none is greater or less than another" (again, the Athanasian Creed) we acknowledge that the Father is greater than the Son and higher in order or hierarchy, as we read in 1st Corinthians 11:3
"the head of Christ is God," and as Christ says in John 14:28
"My Father is greater than I." Paul also says
"Then cometh the end, when he shall have delivered up the kingdom to God, even the Father" in 1 Corinthians 15:24, which shows a sort of hierarchical relationship between the Father and Son. This is one obvious reason why the practice of praying to Jesus or the Holy Spirit has not gained much currency.
Strict Athanasian Trinitarians, however, who hold to that creed, because they believe the Father, Son, and Holy Ghost to be "coequal" and that there is no hierarchy in the Trinity, will have a much higher tendency to pray to each Person of the Trinity independently, especially since the creed says
"we are compelled by the Christian verity to acknowledge every Person by himself to be God and Lord". If you hold to that, you are likely to end up considering it unfair to pray to only one Person of the Trinity, and thus would pray to all three.
Now Oneness Pentecostals, being modalists (that is, believing that God is only one Person, namely Jesus Christ, and that he goes by the names of Father and Holy Ghost sometimes) will not see praying to Jesus or the Holy Ghost as at all different from praying to the Father, since in their mind, all three are one and the same person.