The notable songs confirmed for the Shrek films are Accidentally in Love by Counting Crows and Holding Out for a Hero by Bonnie Tyler from Shrek 2, and Fearless Hero, La Vida Es Una, Por Que te Vas and This Is the End from Puss in Boots: The Last Wish. Those two films are also the only ones with a named soundtrack album: Shrek 2 has Shrek 2: The Motion Picture Soundtrack, and Puss in Boots: The Last Wish (Original Soundtrack) belongs to the 2022 film.
Shrek 2 is the better documented of the two on the music side. It played in competition at Cannes on May 15, 2004, four days before its US release on May 19, 2004, and Euronews notes that the film had two versions of Holding Out for a Hero. That detail matters for anyone hunting a specific recording: the track on the album and the version heard in the film may not be the same performance.
The Last Wish songs come with their performers: Fearless Hero by Antonio Banderas and Heitor Pereira, La Vida Es Una by Karol G, Por Que te Vas by Gaby Moreno and This Is the End by Dan Navarro. The first of those credits pairs the voice of Puss in Boots with the film's composer, a neat link between the lead performance and the score. Banderas has voiced Puss since Shrek 2, and he also leads the 2011 spin-off and The Last Wish. The other three songs come from singers credited by their own names, which gives the Spanish-language titles La Vida Es Una and Por Que te Vas a place in the film's sound that the earlier Puss film does not have on record. In my opinion, that makes The Last Wish the most song-led entry among the spin-offs, though no source ranks them.
What about the first film? No track list for Shrek (2001) has been confirmed by a studio, trade, news or database source, so no songs are named for it. That includes the often-searched All Star and Hallelujah, and the artists people associate with the film, such as Smash Mouth, Dana Glover, Baha Men and Eels, cannot be credited to it without a confirmed list. The composer credit for that film, John Powell, is confirmed, and Box Office Mojo lists it.
The same applies to Shrek the Third and Shrek Forever After, where Harry Gregson-Williams is the credited composer but no notable songs are confirmed. Puss in Boots (2011) has Henry Jackman as composer and no named songs either. For vinyl or CD questions, the two album titles above are the only ones confirmed for the series.
Shrek 5 has a US release date of June 30, 2027, according to Variety Australia, and its music has not been detailed publicly yet. More on the film is at Shrek 5.