We study several new magneto-optical trapping configurations in $^{87}$Rb. These unconventional MOTs all use type-II transitions, where the angular momentum of the ground state is greater than or equal to that of the excited state, and they may use either red-detuned or blue-detuned light. We describe the conditions under which each new MOT forms. The various MOTs exhibit an enormous range of lifetimes, temperatures and density distributions. At the detunings where they are maximized, the lifetimes of the various MOTs vary from 0.1 to 15 s. One MOT forms large ring-like structures with no density at the centre. The temperature in the red-detuned MOTs can be three orders of magnitude higher than in the blue-detuned MOTs. We present measurements of the capture velocity of a blue-detuned MOT, and we study how the loss rate due to ultracold collisions depends on laser intensity and detuning.
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