Straps are not supposed to provide most of the support. If that is the case then you are wearing the wrong bra size. The band provides 80% of the support. The reason so many women have issues with strapless bras is because they are wearing bras that are too big in the band. Straps hide the issue by taking on more support than they are supposed to but in a strapless it becomes incredibly obvious. When I wore 34C I always had problems with strapless but in 30F I no longer do. It is still a little less supportive than a regular bra but not nearly as bad as when I was wearing a 34 band on my 30 inch ribcage.
The band of the bra is supposed to provide 80-90% of the support in the bra. So, for the most part, a strapless bra will stay up on it's own, provided you don't make too many sudden movements. I do not find them comfortable, and quite frankly I have yet to find one big enough to fit me. You end up adjusting the bra a lot because they tend to slide down a bit.
The aren't as supportive, but some of them have extra boning in them to provide this support.
Actually the underwire provides most of its support and the straps help it stay up a lot but with a strapless bra the only differences are that they have removable straps, some small silicone strips around the whole bra to keep it from slipping, and people buy them a little tight (not too tight) so they'll stay up