Actually, "astable" describes a free-running "oscillator", such as the astable mode of a 555 timer that outputs a constant rectangular wave of specified frequency. Most counters, such as the TTL 7490 or 74LS192 use a series of four "bistable" flip flops to do the counting, each bistable flip flop representing one bit, four required for a 4-bit binary or one-digit decade counter.
Your final choice of flip flop modes is the monostable mode that outputs a single pulse upon receiving a trigger input. Otherwise known as a "one-shot", the TTL versions are represented by the 74121 and 74123. One-shots come in retriggerable and non-retriggerable versions and like counters, may be either positive- or negative-edge triggered. In the TTL world ALL ripple counters are negative-edge triggered and ALL synchronous counters are positive-edge triggered. The 74123 one-shot can be configured for either positive- or negative-edge triggering.
Dean