Apparently tinypic.com has been shut down for International users without any warning. Many here use tinypic on the forum and your gallery page descriptions. Be aware that you may see the image, depending on your location, but not everyone will. It is best not to use this popular service from now on.
Beyond ImageShack and Photobucket if you know good and easy image upload hosts that you use and recommend (especially those with no membership required), please leave a link as a comment.
imgur seems to upload slow...almost like it's hanging. Maybe it's just today?
Though it's a brazilian site, you could use uploaddeimagens.com.br, it's very fast. But I prefer imgur.com too =p
I use MyFrogBag, nothing to complain about so far. Files are (supposedly) linkable forever, no ads, no hassle, very little downtime, and even an efficient and nice support service. File formats supported: .fla .swf .jpg .gif .png .bmp; that's a bit restricitve and my only complaint (pdf and psd would be great). Registering recommended, I'm not sure it's required though but why wouldn't one want to access/delete/edit one's files?
Edit:
@ Gabeh: maj.com has moved to majhost.com. Please update your bookmarks and links. ==> I'll never trust these guys! That's the worst a host can do besides erasing your files.
Lovely :D
Are they gonna become some amalgamation with Photobucket or something?
Well, as many of us have seen; Tinypic had serious technical problems lately: pictures being replaced / names re-used the most spectacular... maybe the shut-down has something to do with this...or perhaps Photobucket is just performing some "reorganization"? Anyhoo...don't think I ever used Photobucket - and now I know that I never will.
I'm gonna have to agree with drr. Imgur is easily the best photo hosting site I've ever used. Reliable, trustworthy, and quick.
I never used tinypic, but really, what the BEEP! Why would they do that!? And even worse, why without warning? They could at least have listed it on their website earlier with the letters in size 72 so no-one could ever miss it.
Yeah, imageshack is the way to go in my opinion. Fast and simple.
Whaaaaaaaaaaaaaaaaaaaat?! 
edit:
Ok, this sucks. I use tinypic for pretty much everything! EVERYTHING!!! 
Just registered for http://www.villagephotos.com/ Best I've seen so far except for the usuals(Photobucket, Imageshack) .GIF support
http://www.iaza.com/
http://www.majhost.com/
... That's all that came to my mind.
Photobucket owns Tinypic. Should we trust them again?
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http://myfrogbag.com/ -----never looked back