I'll be purchasing a used Honda Fit in the next year or so. Has anyone heard of particular years (or trim types, etc.) to avoid? Thank you for your advice.
It would help to clarify where you are from. The trim options and generations are different from region to region, and within different countries within a region.
Assuming you are in the US, your choices are
First Gen - 2007 - 2008
Second Gen - 2009 - 2014
Third Gen - 2015 - Present
Within each generation they were only sold in the US as base and Sport (third gen might be different).
The Sport doesn't really add much, except some stuff that makes the aesthetics better, but also slightly more annoying in different ways (though IMO they look better). With the First Gen the sport was the only way to get Cruise control and power doors/windows factory installed.
The second Gen US models lost
refresh seating (with the headrest of the front seat removed, the front seat back can be folded down to form a lounge-style sitting area).. Just an FYI if you had your heart set on that nifty feature. It's back in the 2015s:

Mine was a 2009 - I totaled it so I can't speak for reliability but if you are purchasing on gas mileage look 2011 or newer. I never averaged more than 32 combined mpg on mine. Doesn't seem to be an issue in the 2011 or newer.
We've had a 2008 and a 2009 Sport, both automatic. No problem averaging 38 - 40 MPG with either of of them with mainly highway driving.
The engine and the mileage get better with the newer models, and I think the second generation is a bit better built then the first (I noticed less windnoise, but this could be vehicle specific).
If given the choice I'd match them with a manual, not an automatic.
I'm tempted by the 2015 for the simple reason that refresh seating is back. It's a gimmick, but I love the "magic seats".