Well, postmodernism is a fuzzy concept, and the distinction between modernism and postmodernism is rather blurry; many people consider postmodernism to be an extension of modernism, some refuse to admit it exists (which is pretty postmodern in itself really).
Anyway, 'modernism' in music probably started at the end of the 19th century. It was essentially a rejection of Romanticism and what it represented: fiery, unbridled emotion, hedonism, nationalism, worship of the hero concept, etc. At the time many felt that these ideals could only lead to disaster, and the fears really materialised in the first world war. In the early 20th century, modernism crystalised into several different forms: compare Debussy with Mahler with Schoenberg with Hindemith. The essential features of modernist thought are objectivity, rejection of past formalisms (especially those associated with Romanticism), and an interest in the possibilities provided by the promises of science and technology.
Postmodernism developed towards the middle of the 20th century as, if not an outright rejection of modernism, a modification of modernist thought. Some major features of postmodernism are: a rejection of the idea of artistic 'progress' (which is a kind of argument against the existence of this thread but I'll get to that in a bit), an embracing of moral and cultural relativism (i.e. that there is no difference between 'high' and 'low' art, all art is 'created equal', and anything can be art), and also a rejection of the possibility of objectivity, and the emphasis on science.
The second feature I mentioned, relativism, did apply in some way to modernism as well, as that movement favoured experimentation and adoption of non-western traditions into western music.
However, the other two features are the key differences from modernism. Firstly, Schoenberg developed the 'pantonal' aesthetic (or whatever you want to call it) as a way of taking the progress of modernism to its 'logical conclusion'. This was based on the assumption, of course, that music was heading in one 'direction', towards some kind of goal. He noticed that the rules of tonality were being bent further and further, and decided that the inevitable aim of western music was to do away with tonality altogether.
Postmodernist thought rejects this idea outright: art has not 'aim' or purpose: it's not 'progressing' towards some goal. It simply changes, and there is no reason to believe that 'new' is better than 'old' (there's the relativism coming into it), or that 'future' is better than 'present'.
That's what I meant when I said it was an argument against the thread itself. Music has no 'future', it's not 'headed' in some kind of direction. It will simply continue to evolve and people will continue to contribute their own different ideas to the cultural gene pool. Some ideas will take off, some will die; chances are none of them will ever come close to dominate in the way that the Classical or Romantic or even some of the early Modernist aesthetics did in the past. And that's really probably a healthy thing (although admittedly that's my opinion and I would class myself as some kind of postmodernist myself - there's the old rejection of objectivity for you)
Therefore the artist can essentially do whatever s/he wants. For this reason John Cage is often cited as one of the earliest postmodern composers.
Of course, not all artists/philosophers/members of the general public accept postmodernism. It has essentially found no favour in Germany as far as I am aware, but is more popular in other parts of Europe and in North America. But the world of 'western classical music' (whatever that means) is one area of the arts where it seems to be strong.
So essentially I would answer the question of 'where is classical music headed?' by saying that it is headed everywhere and nowhere, and wherever you want to go.
(Also, obbligato: hi I am new; my name is Tom. I think I like this forum quite a bit. Apologies for the lengthy post but I find it a fascinating subject.)