Ignore what was said about .bmp, the reason that isn't used is due to the fact it's not compressed and thus has a large filesize. The actual reason images get artefacts if the settings aren't high enough is due to the compression utilised on most image uploaders. Tinypic, photobucket, imageshack and etc all compress image sizes which introduces more artefacts to it regardless of filetype.
In fact their are few hosting sites that don't compress images that get uploaded, hence you'll find a lot of artists host their own when they can. However if you are really worried about artefacts being introduced to images try somewhere like majhost.com.
The only other reason for using .gif or .png is transparency.
EDIT: However for the purpose of merely displaying your work using an image compressing host is fine, after all you can always zip the raw images, stick them on a non-compressing host or send them via a messenger program. Thus voiding the concerns with compression.