The best host if you want to embed images in a forum. There are paid options (the $25/yr account is a nice option, check it out if you want) but not needed; their free account is fine. You will never "lose" an image in a post no matter how old it is. Easy code generation feature to embed in a post.
Some people complain that the free account limits image resolution, but for web posts just resize in one of your image programs to 800x600 and upload that. It's "big enough" but won't cause a thread to resize too wide for most browsers. Here at AudioCircle you could go a bit larger, maybe 1200x800. I keep my albums and images "private" in the preferences.
Photobucket was never a good option even in the past, because if you didn't login at least once every six months they gave you the "image not found" banner.
I'll give an example: (I clicked "BBCode", one of your options when you select an image, and simply copied and pasted into this post): This is a 800x600 image I uploaded to imgur more than seven years ago. If you were to search this post sometime in the future this image will still be there.