They have great fruits, but there is one type that just spreads like wildfire and produces nothing in my garden. The fruiting one is a small, single plant without runners, or at least it doesn't look like it has...it grows on a weedy, sunny hillside here. I try to get the fruit before the local rodents do.

You can grow a wild strawberry without runners in your garden. I got the original plant from Bluestone over 18 years ago, and have divided it every year since, to make many. They give more fruit when divided into single plants yearly. What a taste!

Fruiting native wild strawberry does have runners. And whether it is a friend or foe depends on your point of view. The berries are tiny but delicious. The 'wild' strawberry without runners is a European import, not native.
There is a strawberry like plant with yellow flowers which is called the 'barren strawberry.' It doesn't fruit and is not a true strawberry.
The native wild strawberry can be considered a weed but it isn't hard to pull out. Left to its own devices, it does spread.

Hmm. I think I'll thin these (they've formed a dense mat), see if they fruit next year, and if not ... out they go. I'm a pushover for these natives!

I guess the weed I hate is the barren strawberry. It's one of those plants that I just have to pull out because it drives me crazy for some reason. It runs all over everything.

It's been recommended as a ground cover though.