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We present a systematic discussion of supersymmetric solutions of 2D dilatonsupergravity. In particular those solutions which retain at least half of thesupersymmetries are ground states with respect to the bosonic Casimir function(essentially the ADM mass). Nevertheless, by tuning the prepotentialappropriately, black hole solutions may emerge with an arbitrary number ofKilling horizons. The absence of dilatino and gravitino hair is proven.Moreover, the impossibility of supersymmetric dS ground states and ofnonextremal black holes is confirmed, even in the presence of a dilaton. Inthese derivations the knowledge of the general analytic solution of 2D dilatonsupergravity plays an important role. The latter result is addressed in themore general context of gPSMs which have no supergravity interpretation.  Finally it is demonstrated that the inclusion of non-minimally coupledmatter, a step which is already nontrivial by itself, does not change thesefeatures in an essential way.Comment: 30 pages, LaTeX, v2: mayor revision (rearranged title, shortened  abstract, revised introduction, inserted section from appendix to main text,  added subsection with new material on non-SUGRA gPSMs, added clarifying  remarks at some places, updated references); v3: corrected minor misprints,  added note with a new referenc

Supersymmetric black holes in 2D dilaton supergravity: baldness and extremality