Sorry Rhett but I have to favor Bags' side of this arguement...and that's coming from someone who can't stand the band. Though, as far as 'cutting edge' is concerned, I tend to think of them more as being a dull knife in a set of cutting edge cutlery.
I will admit that when the college stations first got ahold of the song "Roxanne", I did like it for about 4 months. Then something went terribly wrong within my head between my ears and my brain and it's been all over with since then. To this day, Sting's voice grates on my nerves so bad that I avoid it as much as possible. And that's not even taking his personality into consideration.
Back then, the Balt./Wash. area was loaded with great radio, some of which had direct lines to the NYC and UK underground. (I know, hard to believe for you more recent arrivals.) We got the song early on (late 70s) and one of it's trademarks for being cutting edge is that it helped to introduce the Raggae or Ska beat into New Wave and Punk music. Prior to that, at least within the US, any kind of Raggae or Ska was generally limited to other kinds of rock or left to it's own genre. The exception would be some of the Ska and Black & White scene that was coming out of the UK but The Police were about the 1st to make it popular in other parts of the world. I might be wrong but I think The Clash came along (to the US college market) just afterwards. So, in those regards alone, you have to recognize them as being 'cutting edge'. Actually, pretty much all of their earlier works had that sound and they were very much credited and known for this.
About 2 years later, their music seeped into the Mainstream, with many thinking that "Roxanne" was a new song, and by then, it wasn't quite as obvious. As what usually happens, those who thought this was a new song didn't realize the importance of their influence within the New Wave and Punk scenes as many others then took those ideas and used it within their own music. By then, it was a crap shoot as to which band heard whom first as to who actually influenced them since those sorts of sounds became widely popular and you could then hear them everywhere. Back then, The Police were actually included within the Punk genre but eventually became disenfranchised as they became more Mainstream and Punk went more hardcore.
With all of that said, I still can't stand them! I always liked Stewart Copeland though but that has much more to do with him once being a member of Curved Air than anything else. Always loved that band.