Anything where the world events can affect each player. In the case of mud, you can render these per player. But in the case of falling rocks, say, that impact players and vehicles, every machine needs to be in sync. Otherwise in player 1's game, player 3 gets knocked off the cliff by a boulder, while in player 3's game the boulder passed close to his left but missed...
In that same avalanche situation, you could process the boulder physics on the server, and for the little pebbles and dust, calculate those as special effects on each console because they don't affect the players.