I usually write pop/rock songs, and have a very strong sensibility for pop structures. But I have always tended to be on the progressive end of that spectrum; I use jazzier chords than most of the "rock guys" I know, love to mess with odd meters, flirt w/ atonality, etc. But I've got material I've written ranging from 2 minute pop ditties to 15 minute prog-rock epics to technical solo bass type pieces (mostly from when I was a younger cat with "something to prove").
Clearly I'm the only player here who has written him or herself a 7 minute bass wankfest....
Anywho, ideally I can start with a melody, or a fragment thereof, or at least a lyric idea, and run with it from there. As I (hopefully) mature as a writer, that's how I'd like to begin every piece I write; I've come to believe the melody to be, under almost any circumstances, the most critical element of most "popular music".
That said, I find that I have very frequently relied on farting around with stock or semi-stock chord structures to start with. Hell, how many rock songs are I-IV-V? I try not to live in that box, but sometimes rock'n'roll compels you....
I've also discovered a great love for improvising with a band and seeing what comes out. You can treat a jam as a sketchpad and develop ideas from the jam into one or more actual songs. Or you can just consider the jam itself a song that can never be repeated.
I was in a band that released a well-received CD of material created in the latter vein back in 2003. We recorded probably ~10-20 hrs of jams total. I mixed them all as played, then our drummer did a surgical editing job to get rid of the stuff that just didn't work, to create some artificial "transitions", etc. It got some pretty amazing reviews, to my surprise.
I guess I ultimately agree with the previous poster who mentioned that the initial inspiration is the trick. Once I've got a musical bit in mind, things will eventually fall into place for a song.