The most common use of tomber is an intransitive verb:

• Je suis tombé (de la chaise). ¹
• Je suis tombé (par terre).²

In Classical French tomber was sometimes used in this sense conjugated with *avoir. It is now outdated and considered “bad” French (at least in France), although some people say it, especially young children who can’t speak properly yet, but they’re always corrected at school (and by parents).

Tomber is correctly used with avoir as a transitive verb, for instance:

• Il faisait tellement chaud que j’ai tombé la veste.³

and in some phrases like:

• Il a tombé son adversaire. ⁴
• Il a tombé le masque.⁵

I have never heard je me suis tombé, although it seems to be a “mistake” some people make, but I have no idea where.

References: TFL, BDL

¹. _{I fell off the chair.}
². _{I fell on the floor.}
³. _{It was so hot I slipped off my jacket.}
⁴. _{He vanquished his opponent.}
⁵. _{He took off his mask.}