For me the thicker the blade the more I tend to convex it. Axes, machetes, choppers and other thick blades really take to a convex edge but some others can benefit as well. The last few edges I converted to convex were a Becker BK-2, a Cold Steel Kukri and an Izula. I don't convert all my knives to convex and maintain some V, hollow, scandi, chisel and asymmetrical edges.
I do almost all convexing with a slack belt sander these days for speed reasons, but I have used wet/dry sandpaper and loaded strop for converting as well as maintenance.