Reddit reviews The Best American Comics 2017 (The Best American Series ®)
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For what you're talking about specifically, you might try looking at the Best American Comics anthologies. Here is the 2017 edition, for example. I'm sure there are great periodicals dedicated only to that, but I don't know—some are probably mentioned in the anthologies. As for short fiction in general (not illustrated), a few literary magazines that are popular these days that are a little varied in flavor include:
Online:
Print:
A lot of different work appears in these sorts of mags, and they're not quite what you're talking about, but if you want to just see a bit more of what contemporary literary short fiction is out there, I suppose these are good places to start.