A lot of people seem to use photobucket, but I prefer not to because: firstly because it takes forever and secondly people can take your username from the URL then browse all your photos. So, here are some sites that don't require you to sign up for any account, you just upload your picture, then copy the URL.
I have personally used all of these, and as far as I know they are all 100% safe to use.
http://imageshack.us/
http://imgur.com/
http://imageftw.com/
http://www.hostthenpost.com/
http://tinypic.com/index.php (this one makes you fill out a captcha before you can host)