What is the biblical basis for switching between the Father, the Son and the Holy Spirit as recipients during prayer?

Biblical pattern, described in my answer below, may differ from actual practice. I have indeed heard people praying in the manner you described, especially among Pentecostals who emphasize the experience of the person of the Holy Spirit. Typically they invoke the 3 persons according to the request:

• Father: for thanksgiving for life, for fulfillment of His promises (like how David prays the psalms)
• Jesus: for everything connected to His ministry while on earth: saving us from sin, healing, deliverance / protection from evil, for miracles
• Holy Spirit: for spiritual presence, for life direction, for empowerment, for gifts like tongues, prophecy, etc.

Biblical pattern

I cannot speak for all denominations, not to mention non-Trinitarian churches like LDS and JW, but the pattern I'm familiar with (Reformed tradition) is well described by a well-cited and well-respected Reformed pastor John Piper in his 2009 answer to the question "Does it matter which Person of the Trinity we pray to?":

So the pattern that you find almost uniformly -- I say almost uniformly -- throughout the New Testament is to pray to the Father in the name of the Son by the power of the Holy Spirit.

Another Reformed pastor Matt Chandler devoted a sermon The Trinity and Christian Prayer to this question as well:

Our entire Christian life is lived BY the Spirit THROUGH the Son TO the Father. That’s going to be the framework for how we think about prayer over the next few minutes. We’re going to think about how prayer is by the Spirit, how prayer is through the Son, and how prayer is to the Father.

The formula by Matt Chandler (BY... THROUGH...TO...) is very similar to how C.S. Lewis viewed the Trinitarian God's role in a Christian's prayer, from the following quote of Chapter 2, Book 4 of his famous book Mere Christianity which probably can speak for all Trinitarian denominations and which helps me a lot in directing my personal prayer:

You may ask, "If we cannot imagine a three-personal Being, what is the good of talking about Him?" Well, there isn't any good talking about Him. The thing that matters is being actually drawn into that three-personal life, and that may begin any time —tonight, if you like.

What I mean is this. An ordinary simple Christian kneels down to say his prayers. He is trying to get into touch with God. But if he is a Christian he knows that what is prompting him to pray is also God: God, so to speak, inside him. But he also knows that all his real knowledge of God comes through Christ, the Man who was God—that Christ is standing beside him, helping him to pray, praying for him. You see what is happening.

God is the thing to which he is praying—the goal he is trying to reach. God is also the thing inside him which is pushing him on—the motive power. God is also the road or bridge along which he is being pushed to that goal. So that the whole threefold life of the three-personal Being is actually going on in that ordinary little bedroom where an ordinary man is saying his prayers. The man is being caught up into the higher kind of life—what I called Zoe or spiritual life: he is being pulled into God, by God, while still remaining himself.

No, but perhaps we need to define prayer in this context. In the OT all prayer was to God - they knew of one God, Yahweh, who is also referred to as Father. In the NT, Jesus came to reveal the Father, his Father, God.

We pray to the Father as Jesus modelled in his prayers. There is every great reason to give praise to Jesus and thanks for his wonderful life and sacrifice. Is this not prayer?

Prayer is no more than appealing to a heavenly power in praise and recognition. In that recognition, we ask for our needs and for God's provision - as Jesus said to do, in Jesus' name.

Praying to the Holy Spirit is really praying to the Father. Clearly shown to be the Spirit of the Father in the gospels.

for the Holy Spirit will teach you in that very hour what you ought to say." Luke 12:12

For it will not be you speaking, but the Spirit of your Father speaking through you. Matthew 10:20

Is there a basis for praying to 3 separate identities? No, not according to Jesus, the son of God - through whom are all things. 1 Cor 11:12, Rom 11:36. We also read that Jesus' very life comes from the Father (John 6:57) and the Father is above all - Jesus included, as the 'way' to the Father. So when we pray to God, it is through Jesus we may come before the Father and not by any other way.