In Mexico, when you want to talk about getting a flat tire, the verb you're likely to hear is poncharse. So if you want to say you got a flat tire, that's where our phrase comes in:
Se me ponchó una llanta
I got a flat tire
If you want to talk about other people who got flat tires, you can say:
Se te ponchó una llanta
You got a flat tire
Se le ponchó una llanta
He/She got a flat tire
Se nos ponchó una llanta
We got a flat tire
Se les ponchó una llanta
They got a flat tire
A flat tire, the tire itself that is, is called a ponchada. And the plural version is ponchadas. You may also hear the word ponchadura.
If you want your tire fixed, you might want to go to a "taller para reperar llantas" - a workshop where they repair tires. These places go by several names:
"ponchaduría", "desponchaduría", "desponchadora", "vulcanizadora" or "vulka".
These workshops aren't your Sears or Wal-Mart tire centers, but instead might look something like this:
In fact, if you take a drive through a Mexican neighborhood, you may see a desponchadora that hopefully isn't as quite as untidy as the one in our photo.
Like always, in the Spanish language there's more than one way to skin a cat. Especially if you're talking about flat tires. Other words to take a look at are pinchar, gomas, and desinfladas to name a few. Wordreference.com is a great place to look if you're interested in alternatives.
Anyway, that's it for today. Be careful not to ponchar any llantas in the near future!