Dead, broken, or hanging limbs are the exception to every timing rule. If a branch is a danger to people, cars, or your roof, have it removed as soon as you notice it.
For regular structural trimming of shade trees like live oaks, the cooler and drier months are generally the easiest on the tree. Growth slows down, cuts have time to close before the heavy summer growth push, and it's easier to see the tree's structure.
Late winter and spring are also the right time to get ahead of storm season. Clearing weak limbs and thinning dense canopies before June means your trees go into hurricane season lighter and better balanced.
Palms are a little different. Remove only fully brown fronds and seed pods; cutting off green fronds weakens the palm and can leave it hungry for nutrients.
Whatever the season, avoid topping. Cutting the top off a tree causes a flush of weak, fast-growing shoots that break easily. Proper pruning cuts back to a branch or the trunk collar, and it keeps the tree healthier for longer.