Just about the entire city is gentrifying, getting massive capital improvements, and getting expensive.
To the above poster, looking at the 5th ave, king ave, northwest puts you in Columbus, outside of UA and grandview. While there are more rentals here, they are getting pricey.
3 cats will definitely be difficult. I wouldn't rent to 3 cats for my units.
East of 71 north of downtown is a HUGE area, and it's not all bad. I'd avoid linden and milo grogan myself, but would have no problem living north of Morse. That area is heavy immigrant and has some really cool international flair right now.
West broad west of downtown is franklinton and hilltop. Franklinton has a massive 500mm capital improvement project going on and the area is out of control for investors trying to buy in. It's in transition, but by no means "avoid at all costs".
If you want to walk/bike to OSU and be close to bars/restaurants, stuff to do, right now you are looking at grandview/tri village area, short north (Victorian village, Italian village, Harrison west), the circles, university district (I would avoid as an adult! But tons of fun if you're in your 20's), weinland park (transitioning area), UA around lane, and southern clintonville.
If you are ok with a commute, but want walkable for other things, you can hit the burbs, or south of downtown. German village is phenomenal, and would be right around bike/car range. Otherwise look around old Dublin, old worthington/northern clintonvillle, downtown Gahanna, or even downtown Powell if you want to be that far.
I'm not a huge fan of olde town east for walkable. While a pretty neighborhood that has been transitioning for what seems like 15 years, I don't see much in the way of things to do outside of franklin park. May be my personal bias.
I would avoid east of parsons avenue in Columbus, and I personally don't like the east and south side too much. While there isn't anything inherently wrong there, it's just not my choice.
Long term to buy with no kids, i would be city of columbus, either short north or north of grandview, south of UA. Avoid those pricey property taxes.
One thing will be in common with all of my recommendations, expensive!
Sent from my iPhone using Tapatalk