The powerful methyl(trifluoromethyl)dioxirane (1b) was employed to achieve the direct oxyfunctionalization of 2,4-didehydroadamantane (5), spiro[cyclopropane-1,2'-adamantane] (9), spiro[2.5]octane (17), and bicyclo[6.1.0]nonane (19). The results are compared with those attained in the analogous oxidation of two alkylcyclopropanes, i.e., n-butylcyclopropane (11) and (3-methyl-butyl)-cyclopropane (14). The product distributions observed for 11 and 14 show that cyclopropyl activation of alpha-C-H bonds largely prevails when no tertiary C-H are present in the open chain in the tether; however, in the oxyfunctionalixation of 14 cyclopropyl activation competes only mildly with hydroxylation at the tertiary C-H. The application of dioxirane 1b to polycyclic alkanes possessing a sufficiently rigid framework (such as 5 and 9) demonstrates the relevance of relative orientation of the cyclopropane moiety with respect to the proximal C-H undergoing oxidation. At one extreme, as observed in the oxidation of rigid spiro compound 9, even bridgehead tertiary C-H's become deactivated by the proximal cyclopropyl moiety laying in the unfavorable "eclipsed" (perpendicular) orientation; at the other end, a cyclopropane moiety constrained in a favorable "bisected" orientation (as for didehydroadamantane 5) can activate an "alpha" methylene CH2 to compete effectively with dioxirane O-insertion into tertiary C-H bonds. Comparison with literature reports describing similar oxidations by dimethyldioxirane (1a) demonstrate that methyl(trifluoromethyl)dioxirane (1b) presents similar selectivity and remarkably superior reactivity.