What is postmodernism? It's a stance and a series of ideas that's really badly defined.
Generally it is agreed that postmodernists reject the idea of ideology and overarching narrative in favour of individual ideas and meta-narratives. They reject ideas like Moral Universalism and Objective Reality. They say that these ideas are results of years of culture rather than an innate human truth. Ultimately, postmodernists are sceptics and tend to be ironic also.
What does this mean for Music? Postmodernism in music is usually manifested as a subversion to modernist ideals. It can also be defined more generally as and ironic, subversive form of music. Lyrics can take the shape of meta-narratives or even in the case of some bands subvert the idea that they need to make sense (see: CAN).
A Reaction to Modernism in Classical Music
Towards the start of the 20th Century modernists like Schoenberg were creating grandiose pieces that were dense with sound, they filled up all available space in the ear. Most were atonal, they didn't sound good to the ear. As a reaction to this in the 60's people like Steve Reich and Philip Glass started to use less instrumentation, less 'music'. They started the Minimalist movement.
An interesting sidebar is that 'The Viking of 6th Avenue' - Moondog was an inspiration for Glass and Reich. With his use of strange time signatures and counterpoint. Perhaps this was the inspiration for Reich's Electric Counterpoint?
Listening to pieces like Electric Counterpoint or Glassworks is a polar opposite to Schoenberg's Peripetie it feels like there's no music but it's very tight and deliberate. Another characteristic is the, while strange sometimes, time signatures the music never fails to have a beat or pulse. Minimalist music is generally progressive or at least evolving over time. New instruments are added one by one mostly, they feel carefully picked to play just their notes with little embellishment.
Another idea that Minimalists like Reich use is phase, most easily demonstrated in Clapping Music, also notable is Pendulum Music which manages to use this idea of phase and also the idea of progressiveness quite well.
Rock and Pop
Ultimately postmodernism's influence in pop and rock can be traced to the New York rock scene to an extent bands like Talking Heads, The Velvet Underground etc. are all cited as being postmodern. But what makes these examples?
Postmodernism embraces the idea that we exist as a result of cultural influences and thus will generally be eclectic. Most postmodern music tends to be in some way ironic or subversive. The music will tend to care less about what is perceived as 'upper' and 'lower' music, that is a street performer is valued at the same level as an orchestra in the concert hall.
Talking Heads' seminal album 'Remain in Light' shows this quite well. It is not afraid to use things like poly-rhythms and a clear African influence throughout the album (see: Listening Wind). A clear departure from music at the time. But it also has a certain feeling of detachment from musical trends whilst being self aware sometimes Talking Heads' music feels almost neurotic and insane but it is aware of it.
Talking Heads isn't the only example of it, most of the same New York rock scene were of a similar calibre and it rubbed off on Eno, Bowie, etc. That's not to say that this group were the only postmodernists creating music Art rock as a genre had taken off in the 70's with bands such as Pink Floyd utilising the idea of a concept album with albums like The Wall and Dark Side of the Moon these albums all tended to flow together and have unifying themes whilst (especially in The Wall) creating a kind of meta-narrative of the time. Art rock helped foster genres like new wave, post-punk and synth-pop all of which remain strong influences on the current canon of music.
As for pop music, there exist many examples of the ironic and sometimes meta-narritive filled songs, even those which made it into the charts. A notable example is Pet Shop Boys' album Very and the single Go West. Pet Shop Boys as a very accessible band using a lot of bubbly synths subverts the pop outlook at the time constantly Go West is almost western propaganda with its music video hammering home that point, it all feels very ironic. The Pet Shop Boys were vehemently against things like rock and roll and tended to reject the celebrity norm by wearing avante-garde clothes and refusing to pose for cameras.
Other Music
A big hitter in the past few years in terms of a 'new' style of music is Vaporwave (and it's one million sub-genres) it is argued whether or not this is truly postmodern or just anti-mainstream. I'd like to imagine that at least initially there was some postmodern influence because of the clearly subversive use of samples to that of modern music (e.g. hip-hop) but whether or not this counts as postmodern is another question. Whilst a lot of vaporwave remains anti-consumerist it's again questionable whether or not this is a nod to postmodernism or not. Clearly vaporwave is essentially replicating 80's and 90's popular culture. But is it doing so in a self-aware enough way to be called postmodernism? This arises because of the mystery surrounding a lot of the creators and the lack of clear knowledge as to it's creation.
Perhaps we are looking at some kind of meta-postmodernist creation made to comment on the state of the postmodern?