a. Singing is a human response to God.
Singing is not a means to impress or appease God or otherwise earn his favour. The people who sing are already God's people, and they are singing in response to what God has done for them. They don't sing in order to become God's people - they sing because they are already his!
b. Singing is not a prescribed act of service.
In the Old Testament, God prescribed the particular way that his people were to worship him in terms of the cultic service at the temple. That service involved priests and sacrifices. But singing was never part of the prescribed acts required at the temple. Singing was a spontaneous, responsive act, not something that is commanded as part of the cultic rituals.
c. Singing occurs in many contexts
Singing certainly occurs at the temple, but it also in other locales, often in response to something that the Lord has done. See, for example, the song of Moses in Exodus 15 and Deborah's song in Judges 5.
d. Singing can be to God and to others
Sometimes singing is addressed to God, and sometimes it is addressed to others, to teach and to exhort. In fact, you could say that there are three modes of singing - exaltation, instruction and exhortation. However, it is very important to note that most songs are a mix of all three.
For example Psalm 138:1-2 could be described as exaltation. But in the verses that follow reasons are given for exalting God in this way, which is moving into instruction.
Psalm 106 is an example of instruction. The bulk of the psalm is a rehearsal of the events of the exodus years - the rebellions, the golden calf, the refusal to enter the promised land. But the psalm still begins and ends with an exhortation to praise the Lord, because his love endures forever, in spite of the unfaithfulness of his people.
Psalm 95 is an exhortation to praise God with singing and to worship him with humble submission and adoration. But this exhortation concludes with instruction regarding what happened when a previous generation did not hear his voice and hardened their hearts against God
The word 'praise' is rightly applied to all three modes: we can praise God directly, we praise God when we 'proclaim the mighty acts of the Lord' (Ps. 106:2), and we praise God by the act of exhorting others to honour him (Ps 96:2-4).
e. Singing can be in the singular or the plural
Some songs use the first person plural (Ps. 44:8; 95:1) and others the first person singular (Ps. 139:14). However, there are two important things to note:
1. Many psalms oscillate between the singular 'I' and the plural 'we' or 'us' (e.g. Psalm 103).
2. 'I' songs can be sung by a group of people. Israel's very first song begins like this (Ex. 15:1-2).
It is a mistake to think that 'I songs were 'individual' songs, only to be sung in private, and 'we' songs were for singing corporately.
f. The variety of songs.
We should note the sheer breadth of what was sung in the Bible - confession, lament, praise, thanksgiving, history lesson, exhortation, prayer for deliverance, prayer of hope, petition.
g. Content and response
Typically a biblical song includes both content and response, telling us something about the Lord, and calling on us to make a response of some sort. There is no song which is 'pure content' - that is, just a recital of information about God. There is always a call to make a response to what has been said about the Lord.
h. The 'new song'
Israel's foundational song in Exodus 15 is the celebration of God's great act of salvation in rescuing his people from Egypt. But this Song of Moses is 'transposed' in Revelation 15:3-4 to mark a new stage in salvation history. It has become 'the song of God's servant Moses and of the Lamb'.
The new song is very different from Exodus 15. There is no horse and rider thrown into the sea, for example - though the core theme of God's great deliverance and the fear of the nations are common to both songs.
Isaiah 42:10 announces that the Lord is about to do a new thing - a new act of salvation, even better that that great act of salvation from Egypt - and because of this, God's people will no longer sing the old song. This perspective is reflected in Revelation 15. The new thing has been done. God has accomplished the greatest and most mighty act of salvation, in and through Jesus, which means that the old song is not our song any more - we must sing the new song about salvation through the Lamb.
We have to ask whether any Old Testament song needs to be transposed into a 'new' key - that is, the 'new covenant key'. Some psalms don't need any adjustment, but others need to be modified to express New Testament realities (e.g. Ps. 27:4-5).