Either one is fine and has it's pluses and minuses. South Side Slopes is harder to navigate and depending where it is, you may be outside of reasonable walking distance to amenities (not to mention huffing up the slope). I believe the slopes is as safe as any other city neighborhood.

Greenfield is more residential with some smaller business districts. Generally easier to navigate, walkable. Much like folks have said 'beware realtors labeling the neighborhood the slopes when it's really Arlington, Mt Oliver, Allentown'...realtors also label parts of Hazelwood as Greenfield. This isn't to discourage, put any judgement on neighborhood titles, etc, but just know realtors play a little loose with neighborhoods to make it sound like it's in a 'nicer one'.

I would keep both options open and take close looks at the houses you're interested that meet your other criteria (bedrooms, yard or not, etc). For houses below ~175k in either neighborhood, I'd be prepared to take a really close look especially if they are 'recently updated' because for that price range you'll see a lot of amateur flips or poor-quality jobs where the upgrade is really to hide something or more make it more presentable. When we were looking for houses ~4 years ago in the 120k price range within city limits, I'd say 9/10 houses either needed significant work or updates (think crumbling dry wall, electrical, no major updates since 1970) or were flip jobs that were either poorly done (wonky tiling, trim work, etc) that would need to be redone and/or was hiding something worse behind it.

If you get to the point of the home inspection, know ahead of time it is largely a visual inspection. Home inspectors usually don't 'peel back' anything or do more than a surface level investigation. The inspection helps withs some things but is not a thorough report of EVERYTHING in the house.