I just finished two back to back trips into the Palisades. Large patches of snow are lingering below the chutes between SW Chute #1 and the U Notch. The snow is firm in the mornings and softens up by mid day. The biggest issue related to the snow patches is they will make you lose and gain elevation as you traverse to Starlight and (especially) North Palisade. Bringing crampons and an ice axe would help with these patches but then you'll be walking on a lot of rock in between the snow. This leaves you with the choice of taking them on and off (which is time consuming) or leaving them on for the rock sections (which makes for a lot of awkward walking).
I wouldn't recommend an ice axe or crampons and would instead allow extra time if you're headed to North Palisade. The one exception is Starlight. If you're planning on doing the class 5 chute you may find them useful for the short approach section. You can then leave them at the base to pick up upon your return. Careful route planning can help you eliminate the need for the crampons though even on this one.
Here are pictures from a couple climbs you might find helpful:
Thunderbolt Peak September 1-2, 2011Starlight Peak August 8-10, 2011Be safe and have a great time up there. If you have any specific questions feel free to get in touch!