my

We don’t traditionally think of my as having synonyms, but there are several situations in which it can replace—or be replaced with—another term.

my own 

The phrase my own is sometimes used as a more emphatic way of saying my, as in I can’t believe I was betrayed by my own family.

mine

Sometimes you can reword what you’re saying to use mine instead of my. For example, instead of saying This is my seat, you could say This seat is mine. 

Mine is also an archaic form of my. This was especially used before a word beginning with a vowel sound, as in mine eyes, or following a noun, as in I must do what is duty mine. 

our

Instead of saying things like This is my house and her house, we use the plural form our, as in This is our house. 

me

Sometimes, my is used where the word me could also be used, such as in sentences like this: Did you hear about my getting promoted? Historically, it was considered more proper to use my before gerunds (e.g., getting in the preceding example), but using me is now more common and often sounds more natural.

Me is sometimes used in place of my in a particularly informal dialectical use in the U.K. and other places, as in I’ll collect me things and be on me way.