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This paper argues that flatness appears as a central notion in the verification of counter automata. A counter automaton is called flat when its control graph can be ``replaced'', equivalently w.r.t. reachability, by another one with no nested loops. From a practical view point, we show that flatness is a necessary and sufficient condition for termination of accelerated symbolic model checking, a generic semi-algorithmic technique implemented in successful tools like FAST, LASH or TReX. From a theoretical view point, we prove that many known semilinear subclasses of counter automata are flat: reversal bounded counter machines, lossy vector addition systems with states, reversible Petri nets, persistent and conflict-free Petri nets, etc. Hence, for these subclasses, the semilinear reachability set can be computed using a emph{uniform} accelerated symbolic procedure (whereas previous algorithms were specifically designed for each subclass).