Schur Q-functions were originally introduced by Schur in relation to projective representations of the symmetric group and they can be defined combinatorially in terms of shifted tableaux. In this paper we describe planar decompositions of shifted tableaux into strips and use the shapes of these strips to generate pfaffians and determinants that are equal to Schur Q-functions. As special cases we obtain the classical pfaffian associated with Schur Q-functions, a pfaffian for skew Q-functions due to Jozefiak and Pragacz, and some determinantal expressions of Okada. We also obtain results for Schur P-functions, results for supersymmetric Schur functions, and generalizations to variable sets subscripted by arbitrarily ordered alphabets.