We have found several new spontaneous fission (SF) activities and suggested assignments for some of them to rutherfordium (element 104) isotopes. Their half-lives and production cross sections have been measured by collecting recoils on a moving nickel or steel tape and transporting them past stationary mica track detectors at known velocities. The following tentative assignments are based on several cross bombardments and comparisons between experimental and calculated production cross sections: $^{256}\mathrm{Rf}$(9\ifmmode\pm\else\textpm\fi{}2 ms), $^{257}\mathrm{Rf}$(3.8\ifmmode\pm\else\textpm\fi{}0.8 s, 14\ifmmode\pm\else\textpm\fi{}9% SF), $^{258}\mathrm{Rf}$(13\ifmmode\pm\else\textpm\fi{}3 ms), $^{259}\mathrm{Rf}$(3.4\ifmmode\pm\else\textpm\fi{}1.7 s, 9\ifmmode\pm\else\textpm\fi{}3% SF), $^{260}\mathrm{Rf}$(21\ifmmode\pm\else\textpm\fi{}1 ms), and $^{262}\mathrm{Rf}$(47\ifmmode\pm\else\textpm\fi{}5 ms). Presently we are unable to assign several other new spontaneous fission activities with half-lives of \ensuremath{\sim}15 ms, \ensuremath{\sim}22 ms, \ensuremath{\sim}100 ms (Zl104), \ensuremath{\sim}1.6 s, \ensuremath{\sim}5 s, \ensuremath{\sim}5 s, \ensuremath{\sim}30 s, and \ensuremath{\sim}47 s. Contrary to other observations, we have not found any indication for an 80-ms spontaneous fission activity, earlier credited to the spontaneous fission decay of $^{260}\mathrm{Rf}$, in reactions in which we expected to produce this isotope. Our measurements support a shift in the spontaneous fission half-life systematics at element 104, first proposed by Flerov and Oganessian et al. and later predicted by Randrup et al. and Baran et al. and attributed to the disappearance of the second hump of the double-humped fission barrier and a weakening of the 152-neutron subshell.