I use table sugar to prime many beers. Belgians, a few IPA, and some older American Styles from the pre prohibition days, like my 1909 Kentucky Common recipe.
I think in some beers it makes for tinier, tighter bubbles and a nice head.
For most of my other beers, English ales and others, I tend to use basic corn sugar.
You can use any fermentable for priming. I give detailed info on priming with alternative primers including fruit juice (including the link to the podcast,) and other sugars in my bottling stickey- Scroll to the lower half of
this post.
I've used date palm mollasses and brown sugar before, the mollassess was a bit too intense, though other folks liked the beer. Brown is great with pumpkin ales.
Most priming calculators give the amounts for both table sugar and corn sugar.