In December of '09 I was caught smoking pot outside. I was never arrested, but given a "misdemeanor citation." I had to go to court and was put on probation. I live in Tennessee. I am applying for an apartment and they run a background check. Would this show up, since I was never arrested?

A misdemeanor citation IS an arrest. You could have been taken to jail or given a citation to appear in Court at a later date.

You can't be put on probation unless you are arrested and either found guilty or plead guilty.

I do background checks - in NY. This WOULD appear in NY.