There are several Psalms that are considered "messianic" by the majority of Christians. These are the Psalms that the New Testament explicitly quotes with regard to Christ: Psalms 2, 8, 16, 22, 40, 45, 69, 102, and 110 are examples. Quotes from the Psalms are scattered all through the Gospels and Hebrews as a demonstration of who Jesus Christ was, and would have been a powerful argument to the Jewish people at the time. No Christian can seriously dispute that at least some Psalms are Messianic, but I've come to believe that the vast majority---perhaps even all that Psalms are Messianic.
Certainly all the Psalms are Messianic to some degree or another simply by virtue of the fact that the Psalms reflect the experiences of God's people on earth. Christ, of course, was the ultimate example of God's Man on earth, so there is a sense where He ultimately fulfilled the Psalms. Now, some heresies and some blasphemies (and probably some blasphemous heresies) have been developed from this very idea. We want to be very careful to maintain Christ's Impeccability (i.e. He did not sin, and He could not sin); so when we go to a passage like Psalm 51, we want to be sure we don't think it indicates that the Lord Jesus was ever separated from the Father for sins He committed. On the other hand, we are positively told that He was made sin for us (although He Himself knew no sin), and so we can fairly say the He understands the full weight of sin. In fact, Christ knows the weight of sin much better than any of us ever will: He bore our sins in His body on the Cross, He was made sin for us. This is a very intimate thing. None of us will ever truly understand just what that was like: we'll never know sin the way He did.
But even if we concede that not all the Psalms can be truly applied to the Lord Jesus, I think we fall far short of understanding the full extent to which they contain God's thoughts about the Son. So I'd like to think just a little bit about Christ in the Psalms. Perhaps this is more of an "application" and less of an "interpretation", but I really think it's valid.
Let's start with the Chief Musician. This title is used in 55 Psalms as well as the last chapter of Habbakuk. The Chief Musician is the Lord Jesus Himself, and every Psalm addressed to the Chief Musician is to some degree Messianic. Certainly several of the Psalms generally-accepted to be Messianic are addressed to the Chief Musician (Psalms 22, 40, 45, and 69 for example); but Hebrews goes on to describe the Lord Jesus as the Chief Musician, quoting Psalm 22:
11 For both he that sanctifies and those sanctified are all of one; for which cause he is not ashamed to call them brethren, 12 saying, I will declare thy name to my brethren; in the midst of the assembly will I sing thy praises. 13 And again, I will trust in him. And again, Behold, I and the children which God has given me. (Hebrews 2:11--13, JND)The Lord Jesus is the one who leads praise in the assembly: He is the Chief Musician.
Another significant title in the Psalms is "Son of Man." This title is used of three people in Scripture: Daniel, Ezekiel, and Christ. Christ alone is referred to as "the Son of Man" with the definite article, and He refers to Himself that way some 200 times in the Gospels. The Psalms are full of "the Son of Man." Hebrews 1 & 2 would remind us of Psalm 8 and the question, "Lord, what is man that thou art mindful of him? or the son of man..." And while in context Psalm 8 doesn't seem to be very Messianic, Hebrews 2:6--10 insists that's exactly what it is.
The Son of Man figures in Psalm 80 (another Psalm to the Chief Musician), which I consider to be significant in light of the parallels between Psalm 80 and Ezekiel. Just like Ezekiel, Psalm 80 is to the One who sits between the cherubim (Ez. 1), it's addressed to the "Shepherd of Israel" (Ez. 34), and discusses Israel as "the vine" (Ez. 15). Psalm 80:17 talks about the Son of Man, who is the man of God's right hand.
And even the Psalms differentiate between "a son of man" and "the Son of Man". Consider Psalm 146: we're not to put our confidence in "a son of man", although Psalm 80 tells us that "the son of man" is the One whom God has made strong for Himself.
One title I've often considered in the Psalms is "the man". I think this is also a prophetic title of Christ, although I haven't fully studied it out. Certainly the Epistles use "the Man" as a title of Christ: there is one God and one Mediator between God and man, the Man Christ Jesus (1 Timothy 2:5). Even Pilate said, "Behold the Man!" (John 19:5). I've yet to find the title "the man" in the Psalms where it isn't applicable to Christ. There are places where "man" or "a man" is used that clearly aren't the Lord Jesus, but "the man" seems to pretty consistently apply to Him.
An interesting Psalm in this regard is Psalm 32:
1 Blessed is he whose transgression is forgiven, whose sin is covered! 2 Blessed is the man unto whom Jehovah reckoneth not iniquity, and in whose spirit there is no guile! (Psalm 32:1--2, JND)The first verse talks about someone who has sin to cover, the second verse talks about "the man" to whom the Lord does not reckon iniquity. Now, I have no doubt that these two verses are really discussing the whole issue of forgiveness: Romans 4 makes that plain. But I can't help but notice that the phrase "the man" is used in the second verse but not the first. Again, I'm not claiming Psalm 32 is Messianic (Christ had no sins to cover), but that the Scripture doesn't use the title "the man" except where it's possible to apply those statements to Christ.
Along those lines, I have come to believe Psalm 1 is Messianic. It is the Psalm about the Blessed Man, and it describes the life of the Lord Jesus. In fact, I think the Gospel is displayed in vv. 5 & 6. For the first three verses, there is one man who is walking godly, but in the last two verses, there is a whole congregation of the righteous. This is the essence of the Gospel: one righteous Man has stood in the place of countless of unrighteous men, and God has accepted Him on our behalf.
But I hasten to say again, lest I'm charged with eisegesis, that every Psalm is to some degree Messianic, as Christ is the ultimate Antetype of the godly man on earth. So in that sense, even if you don't believe Psalm 1 is prophetic, you certainly must admit that there was no more blessed man who walked the earth than our Lord Jesus.
Even Psalm 23 is ultimately Messianic: it is the song of the Lamb. Doubtless the Lord Jesus is the Good Shepherd, and what Psalm 23 says about the Shepherd certainly applies to Him. But He is the Lamb of God, and we can see in Psalm 23 a hint of the confidence He had in the Father while He was walking down here.
One place I've found the Psalms very helpful in Bible study is in tying together lines of truth. That sounds more "brethren" than I meant it to... what I mean is that the Psalms will frequently connect different images from the Old Testament into a single thought. So when I go back and examine those images in the Law or the prophets, I find the connections made between the different images combine into single thoughts. I suppose a good example is Psalm 80, which combines the Shepherd, the Vine, and the Son of Man together. Or we might consider Psalm 84, which combines the sparrow with the Altar. That just screams Psalm 102, doesn't it?
I'm no expert in the Psalms, but I find myself gravitating more and more to them as a series of prophetic images of the Saviour.