When selecting a strapless bra-- you can keep most of the convertible bras, as they only convert to confusion and strange bulges-- be sure it has enough grip around the lower band to stay in place and not transform itself into a belt the next time you race to catch a taxi. You also need to be sure that the bra fits well and does not cause discomfort.
Before you spend good money on a strapless bra, do a little dance in the fitting room and see if it stays in place and doesn't drop down creating the double-decker boob look. See if it doesn't create any weird outlines under your top. Keep moving around a bit to be sure. You will probably need to try on a few as everyone is built differently. If you have V-like torso narrowing to a small waist, longline bras have a better chance to stay in place.