  The problem of determining the maximum number of maximal independent sets in
certain graph classes dates back to a paper of Miller and Muller and a question
of Erd\H{o}s and Moser from the 1960s. The minimum was always considered to be
less interesting due to simple examples such as stars. In this paper we show
that the problem becomes interesting when restricted to twin-free graphs, where
no two vertices have the same open neighbourhood. We consider the question for
arbitrary graphs, bipartite graphs and trees. The minimum number of maximal
independent sets turns out to be logarithmic in the number of vertices for
arbitrary graphs, linear for bipartite graphs and exponential for trees. In the
latter case, the minimum and the extremal graphs have been determined earlier
by Taletski\u{\i} and Malyshev, but we present a shorter proof.
