Learn Russian: У велосипеда сломались оба колеса.

Оба is the form used for masculine and neuter nouns, while обе is strictly for feminine nouns. Since the base word for wheel is колесо, which is neuter, it takes оба.

What case is колеса in, and why isn't it plural if there are two wheels?

It is in the genitive singular. Just like the numbers 2, 3, and 4, the quantifier оба forces the following masculine or neuter noun into the genitive singular. Be careful not to confuse it with the nominative plural колёса — the spelling looks similar, but the pronunciation and stress are different.

If колеса is in the singular case, why is the verb сломались in the plural form?

Even though the noun is grammatically forced into the genitive singular by оба, the subject as a whole (both wheels) is logically plural. Because it represents multiple objects doing the action, the verb takes the plural form.

Why does the sentence start with У велосипеда?

This is the standard Russian way to express possession, literally translating to At the bicycle, both wheels broke. While you could say оба колеса велосипеда (both wheels of the bicycle), using у + genitive is a very natural way to show that the broken parts belong to the bike.