This may seem an unusual topic for this blog to write about. I am thinking about starting an additional blog some time in the future with a more decidedly "spiritual" emphasis. But for now I'll write about the question because I think it is something that some people (including myself) have wondered about. Wayne Grudem addresses the question in his book, Systematic Theology (Zondervan, 1995), pp. 380-381. I find his response fairly satisfying.
A survey of the prayers of the New Testament indicates that they are usually addressed neither to God the Son nor to the Holy Spirit, but to God the Father. Yet a mere count of such prayers may be misleading,
for the majority of the prayers we have recorded in the New Testament are those of Jesus himself, who constantly prayed to God the Father, but of course did not pray to himself as God the Son. Moreover, in the Old Testament, the trinitarian nature of God was not so clearly revealed, and it is not surprising that we do not find much evidence of prayer addressed directly to God the Son or God the Holy Spirit before the time of Christ.
Though there is a clear pattern of prayer directly to God the Father through the Son (Matt. 6:9; John 16:23; Eph. 5:20) there are indications that prayer spoken directly to Jesus is also appropriate. The fact that it was Jesus himself who appointed all of the other apostles, suggests that the prayer in Acts 1:24 is addressed to him: "Lord, who knows the hearts of all men, show which one of these two you have chosen..." The dying Stephen prays, "Lord Jesus, receive my spirit" (Acts 7:59). The conversation between Ananias and "the Lord" in Acts 9:10-16 is with Jesus, because in verse 17 Ananias tells Saul, "The Lord Jesus has sent me that you may regain your sight." The prayer, "Our Lord, come!" (1 Cor. 16:22) is addressed to Jesus, as is the prayer in Revelation 22:20, "Come, Lord Jesus!" And Paul also prayed to "the Lord" in 2 Corinthians 12:8 concerning his thorn in the flesh.
Moreover, the fact that Jesus is "a merciful and faithful high priest" (Heb. 2:17) who is able to "sympathize with our weaknesses" (Heb. 4:15), is viewed as an encouragement to us to come boldly before the "throne of grace" in prayer "that we may receive mercy and find grace to help in time of need" (Heb. 4:1). These verses must give us encouragement to come directly to Jesus in prayer, expecting that he will sympathize with our weaknesses as we pray.
There is therefore clear enough scriptural warrant to encourage us to pray not only to God the Father (which seems to be the primary pattern, and certainly follows the example that Jesus taught us in the Lord's Prayer), but also to pray directly to God the Son, our Lord Jesus Christ. Both are correct, and we may pray either to the Father or to the Son.
But should we pray to the Holy Spirit? Though no prayers directly addressed to the Holy Spirit are recorded in the New Testament, there is nothing that would forbid such prayer, for the Holy Spirit, like the Father and the Son, is fully God and is worthy of prayer and is powerful to answer our prayers. (Note also Ezekiel's invitation to the "breath" or "spirit" in Ezek. 37:9.) To say that we cannot pray to the Holy Spirit is really saying that we cannot talk to him or relate to him personally, which hardly seems right. He also relates to us in a personal way since he is a "Comforter" or "Counselor" (John 14:16,26), believers "know him" (John 14:17), and he teaches us (cf. John 14:26), bears witness to us that we are children of God (Rom. 8:16), and can be grieved by our sin (Eph. 4:30). Moreover, the Holy Spirit exercises personal volition in the distribution of spiritual gifts, for he "continually distributes to each one individually to each one as he wills" (1 Cor. 12:11, author's translation). Therefore, it does not seem wrong to pray directly to the Holy Spirit at times, particularly when we are asking him to do something that relates to his special areas of ministry or responsibility. In fact, through the history of the church several well-used hymns have been prayers to the Holy Spirit [e.g. "Come, O Creator Spirit"- anon, Tenth Century; "Spirit of God, Descend Upon my Heart"- author: George Croly, 1854; "Come, O Come Thou Quickening Spirit" - author: Heinrich Held, 1664]. But this is not the New Testament pattern, and it should not become the dominant emphasis in our prayer life.