There seems to be two ways to identify the Trinity in the Old Testament. By the names used to refer to God or by the special quality of the operations that God seems to be doing in the context. Although in general some names seem to refer to the different persons of the trinity, it does not seem, from what I have read on the subject that one can strictly say, ‘Wherever this name is used it means the Holy Spirit, or wherever this name is used it always means the Son.’ In stead each name is a proper name of the’ One true God’. When the ‘name’ is placed within a context indicating the peculiar operations of one person in the Trinity, than it can be said to refer directly to that person. Even if the name generally refers to the Father, under a given context is may actually refer to the Son:
For to us a child is born, to us a son is given, and the government will be on his shoulders. And he will be called Wonderful Counselor, Mighty God, Everlasting Father, Prince of Peace. (NIV Isaiah 9:6)
I will not speak directly about each name, to try and assign which ones are more frequently used for which person, or if they are at all, as that would be a thesis statement for a doctorate in theology. I will only mention that, in addition to the title of the Messiah which we understand points to Christ, the one name of God that most commonly is assigned to Christ is 'Jehovah'. But as in the quote above and other references, the following 'name samples' appear to be used for Jesus Christ specifically on occasion: Wonderful Counselor, Mighty God, Everlasting Father, Prince of Peace, Adonai, (Psalm 110:1),Elohim, (Psalm 45:6,Hebrews 1:8)
The main subject to consider is not the specific names but the operations of God and how each person of the Trinity has distinct operation in the relation to the other persons. When the Old Testament refers to ‘God’ in general, then no particular person of the trinity is being identified because all three persons of the Trinity are undivided in their operations and roles. The Father, Son and Holy Spirit as acting by the same exact and undivided will, wisdom, power. All three are the author of the operation. God is generally referred to in this way because of the perfect Unity of all three persons with same essence, likeness and divine nature. However, even when there is no distinct mention of a given person in the Trinity by name, because each person has a distinction from the other two and therefore a relation to them, each can be said to distinctively operate where ever God is referred to as operating. This existence, and mutual relation between each person within the single existence of God has been called their ‘subsistence’. Each person has an eminent distinction and relation to the others in their subsistence. When any special impression is made of the special property of any person on any work; then the scripture is drawing attention to that work as assigned peculiarly to that person. Not that any person is passing the ‘baton’ as it were to the other and no longer acting, because all three are always joined in One performing it, but outwardly and as described in scripture special attention is given to one person of the Trinity at special instances.
As we collect the many places in the Bible where particular persons are mentioned under the acts that are being performed, or specifically and directly by their name, as is often the case in the New Testament, we can recognize patterns, or characteristic properties, to look out for. For example I gleaned these characteristics from John Owen's Discourse on the Holy Spirit, in the third volume of His works, which I read around ten years ago and it still impacts me.
Father: The fountain of the Godhead, beginning of divine operations, power and authority, sending
Son: Eternally born of the Father, proceeding from the Father, condescension, grace and wisdom of the father, subsisting, establishing, upholding, making a consistency, making a permanency.
Spirit: Proceeding from the Father and the Son, condescension, immediate works, concluding, finishing, immediately enabling, completing and perfecting acts and mysterious concluding works.
Note: I have not tried to collect Bible versus that illustrates these characteristics because anyone familiar with the Bible will recognize them. However, just referring to major events like the Father 'sending' the Son to be 'incarnate' and the Spirit 'enabling' the Son to 'establish' the work of atonement and 'send' the Spirit to 'perfect' Christ’s holiness in us, gives a big window into the truth of the matter.