Tinypic allows you to upload images up to 200+KB in size (I forget the exact size limit, but you can fit a full-screen JPEG at decent quality), and gives you back both a web page and a short URL to the bare image. There's no registration required, and the clean, bare-bones interface is compatible with all browsers. The server occasionally goes slightly wonky, and there's no telling how long it will continue to exist, but it's perfect for applications like blogs, where reliability isn't a big deal. It seems too good to be true, but I have yet to find the catch.
Of course, they offer no privacy for uploaded images either, as their random image page makes clear, but that's hardly a drawback given the target application.
I am now using Tinypic to host a portrait in the left sidebar. Perhaps showing my face will encourage me to cap the vitriol, or at least to fling it about less often, thereby helping me live up to my New Year's resolution to exhibit more generosity of spirit and less petulance here. Most of the other resolutions aren't panning out, so I might as well try for this one. Sigh.