Emptiness check is a key operation in the automata-theoretic approach to LTL verification. However, it is usually done on Büchi automata with a single acceptance condition. We review existing on-the-fly emptiness-check algorithms for generalized Büchi automata (i.e., with multiple acceptance conditions) and show how they compete favorably with emptiness-checks for degeneralized automata, especially in presence of weak fairness assumptions. We also introduce a new emptiness-check algorithm, some heuristics to improve existing checks, and propose algorithms to compute accepting runs in the case of multiple acceptance conditions.