I've been regularly in Verbier since 2001, except for 2007, and I went three times, always from Savoleyres, up the Pierre Avoi. It's a very pretty route, and the view straight down to the Rhône valley has always impressed me. As far as I know, the name "Pierre avoi" means "rock with a view", and that one certainly has from the summit. The last part is very well secured, first with a ladder, then with metal bars. Steps are hewn into the rock. Nevertheless, one should be a bit careful; the rock tends to be slippery in certain parts.
A nice way to go down is along the Crête de la Marlena, where you reach a completely untouched grass plateau on 2300 meters. There's a saddle to the south and one to the west. Both are like "windows", the southern one giving way to the view to Grand Combin and the western one to the Aiguilles Suisses of the Mont Blanc range (see also the photos I attached to this mountain page).
I climbed Pierre Avoi from Savoleyres on the normal route. The last meters are good secured with two scales and cables.