Depends on what age and model you're talking about. The current car (from 2013) uses a CVT as did the previous gen (2009-2013) both of which last well but don't suit spirited driving but the earlier Corolla Verso (2004-2009) I believe used the horrible MMT (Multi Mode Transmission) automated manual box which was pretty rubbish and prone to problems while the older Avensis Verso (from 2001) used a conventional 4sp torque converter box.
It was, wasn't it. It came near to what we'd now call a crossover, and might have sold better now than it did then. I suppose the RAV-4 took over that 'slot', as newer RAVs are quite a lot bigger than the original.