My favorite food is noodles. This one is correct. “Food” is singular so you need to use “is” not “are”. “My favorite food is hamburger.” This is technically correct, but people would usually say “hamburgers” in this sentence.

The subject of the sentence is "food." "Food" is singular. So the correct verb form is "is," in all three cases.

my favorite food is

They are correct, although the 3rd one should be "hamburgers" but it is still correct.