Personally I don’t agree with these suggestions, for three reasons. Firstly, what we should always ask regarding any biblical statement is, What would these terms have meant to Paul’s original readers and was there anything available at that time that was called by these names? There was a collection of praise items at that time which was called by these names, and that is the Book of Psalms. When Mark says that Jesus and his disciples sang a hymn on the night of the first Lord’s Supper (Mark 14:26), the reference is to the psalms that were sung at Passover time. Many of the titles of individuals psalms say they are songs.
Secondly, in the Greek of this clause, the word ‘spiritual’ probably governs psalms and hymns as well as songs. The term ‘spiritual’ means ‘from the Spirit’, and refers to songs that have been divinely inspired. Therefore, Paul here is advocating that they use inspired material in praise offered in their public meetings.
Thirdly, in a parallel passage in Colossians 3:16 (Let the word of Christ dwell in you richly in all wisdom; teaching and admonishing one another in psalms and hymns and spiritual songs, singing with grace in your hearts to the Lord), Paul equates these songs with the word of Christ and as a suitable and authoritative basis for teaching and admonishing. Only biblical material has this authority.
So Paul is telling his readers that when they come together for public worship they should sing from the Book of Psalms. The impression is given today that the reason why we believe in exclusive psalmody is merely traditional. But that is not why we limit our public praise to the psalms. The reason why we do so is because we believe the Bible requires it.