They do make ska punk/pop punk songs.
It depends on if the song has a guitar off beat or not.
I can't post links yet, so:
Less Than Jake - Liquor Store Has an off beat. But like most ska punk not at all parts of the song.
Less Than Jake - My very own flag is pop punk with horns.

I love Fishbone, The Specials, Operation Ivy and English Beat..

EDIT:

btw, this is something I should've posted earlier...

*Riddims

reggae & Ska genres
 

hi all this is my 1st post been looking around at a few posts and was quite shocked at what people class as Ska. I am a reggae & ska dj Im 56 years old and was listening to this music from the age of about 13 while all my friends were listening to daivd bowie and rod stewart and the likes (both of which i like just incase anyone thought i was putting them down lol). I see bands like The Flaming Tsunamis against all authority etc etc in no way shape or form are these ska bands, their not even Oi! bands, they are for want of a better description punk metal. My collection is seperated into genres as I play pretty much all SKA including some 3rd wave. So I have Jamaican ska which is the original with bands like Skatalites the Maytones The Maytals hippy boys etc. then comes SKA Revival with bands like Specials Selecter Bad Manners etc.
and 3rd wave with bands like the dualers Kales Kaviar etc. I dont play anything outside of this like Oi! because its not ska. simple to remember if you are not aware Ska is for skinheads (and Mods) Oi! is for boneheads and there is as much difference between skins and bones as there is between ska and Oi!.
There are so many reggae & ska genres now its hard to keep up. I do play some newer reggae as well like Tarrus riley Romain virgo Chris martin theres some great new reggae out there at the moment.

I hope my 1st post was'nt to negative I was just suprised by some of the band names i saw lol.