Tejano Mart. Yes–that’s the actual name of the store.
Where I’m from we called it the ‘corner shop’ as they were often on the corners of buildings, but now I call it ‘the shop’ and assume those who need to know exactly which shop either will make an educated guess or ask.
Very Pricey?
Where I grew up, 7-11s were as ubiquitous as Dunkin Donuts are here in New England. As such I carried on saying “7-11” for a while when I moved here, but given I only know where 3 7-11s are now, I eventually stopped. Now I say “CVS”. 
Circle K. That’s the actual name. Ubiquitous in Arizona.
“Bodega”? In Toronto? I thought that was more of a New York term; I’ve never heard anyone use it here. We always call them “convenience stores”. In previous years, when they were more dairy-oriented, we called them “milk stores”. Many of the older chains’ names and logos reflected that: Mac’s Milk, etc.
We had “Herbie’s.” The original Herbie had sold the place three or four owners back, but it was still called Herbie’s.
Then they closed and 7-11 moved in, so now it’s 7-11.
Corner store or bodega.
I don’t find it amazing. But then, I tend to use the store’s name.
There are exceptions. The ‘corner store’ is Bay Center Market. I call it ‘the market’. This is confusing to the SO because the supermarket is called The Market. So I sometimes call it the corner market. The nearest gas station had been Toad’s since I moved here. In the last year or so, after it was sold, its name changed to three letters. I call it ‘the former Toad’s’.
In general though, I call AM/PM ‘AM/PM’, 7-11 ‘7-11’, and so on. As a class, I call them ‘convenience stores’. But I don’t say ‘I’m going to the convenience store.’ I say ‘I’m going to 7-11’ or whatever.
Note: Bay Center Market is just a little too large for me to think of it as a ‘convenience store’, and offers a little too much (like videos, fishing licenses, and a mail drop box) for me to think of it as a ‘grocery store’.
I used to call them “Stop-N-Rob”, a pun on the now-defunct chain “Stop-N-Go”. Now I call them “Kwik-E-Mart”.
Milk bar, except when I lived opposite one and we’d just say we were going over to the shop. It did expand beyond being what I’d call a milk bar when they added petrol browsers and extended until they were the size of a small supermarket. Incidentally, it was (and still is) known to everyone in town as the Wog shop. Judge away.
Huh, I never would’ve equated party stores with convenience stores. Not saying you’re wrong, just I’ve always had them separate in my mind. One you go to for slurpees and the other you go to for pizza slices.
Corner shop.
We have seven within a three minute walk (well, there are more in the other direction, but I don’t really use them), so they all have nicknames too - Nice Shop, New Nice Shop, ShopThatSells Everything, OrganicShop (they happen to have a lot of organic goods), Turkish Shop, School Shop.
All small independent general stories with very minor differences in what they sell. There’s a supermarket, too, and a fruit shop, plus all the other shops more than 3 minutes’ walk away - I have no idea how our area apparently supports so many shops.
Today someone wanted something specific and asked me what the name was of the shop I suggested to them. They have [names*?
I don’t go into those little stores, but hubby does. We call them little stores.
Jiffy Pig.
(Generic for all the convenience stores.)
Same here. I used to have a White Hen right on my block. They all got taken over by 7-Eleven years ago, but I still usually call it the White Hen out of habit.
When I lived in Madison, they had “Open Pantry” stores, which of course everyone called “Open Panties.”
I would speculate that there were more than a few jokes quoting this world-famous line