It's not hard to see people when living in NYC. I've seen some of the most random celebrities at restaurants or on the street. Mariah, Josh Hartnett (many times at Equinox), Keifer Sutherland, Tyra Banks, Joan Rivers, Donald Trump, and a bunch of others.
It kind of pisses me off that I have yet to even see Gaga in the streets. The closest I've ever gotten to her was when she was performing at a club in Madrid a block away from me and I didn't even find out till the next day.
As far as meeting, I've met Rihanna at Barneys during the GGGB era. She was with Chris Brown at the time but he was doing this weird thing where he was running around the store with his friends like a child.
Anyway, it was near closing time and she walked in with her entourage and this was around the time I still liked her and I broke my personal code which is not to 'bother' celebrities.
Anyway, I lurked around the area she was sitting in trying on shoes and her bodyguard asked me if I wanted to say hi so I did and there was this awkward moment where everyone stared back at us and I'm like hi and giggle nervously and she says hi back and then I say something along the lines "sorry, I'm just really nervous" or w/e and she gives me a half smile, so I said thank you and left. She's definitely nicer to her fans now than before because I got this snobby vibe from her at the time.
Fun facts, she has big feet and her hands weren't too soft. I hope now that she's richer, she's using some La Mer on them.
Other major celebs I've met have been through internships and work so it was more professional than me fangirling.