I'm going to say anything that Brian K Vaughn writes, which would have Saga in it (but its kind of a space fantasy... story). Y:The Last Man is an all time favorite. All of the men on earth die, except one.
Preacher is also amazing, real world, but a preacher in Texas is given the Word Of God. So, that's fun. It's complete (so is Y, but Saga is on going).
Sweet Tooth is a great story. Post Apocalyptic.. with animal people (mutated, not anthropomorphic).
Descender is new (and on going) that is really good. Space story, about an android boy living in a world where androids are illegal.
Locke And Key (completed) is amazing. Real world, crazy stuff happening. I don't think its horror, but some might.
We3 is very short, but so so good.
Death Vigil is super great, funny, too.
While you don't like Fantasy, you might still dig Rat Queens. Super fun and funny.
Uhh.. yea. That's a good starter list I think, with a good variety. Check some out and let me know wha you think, I might be able to narrow down some recommendations based on that.
Sorry this was kind of long, I just love comics!
-Edit- I for got to mention, none of these are about supers. I don't typically read about supers either (except Xmen).