We make a model-independent analysis of all available data that indicate neutrino oscillations. Using probability diagrams, we confirm that a mass spectrum with two nearly degenerate pairs of neutrinos separated by a mass gap of Ӎ1 eV is preferred over a spectrum with one mass eigenstate separated from the others. We derive some new relations among the four-neutrino mixing matrix elements. We design four-neutrino mass matrices with three active neutrinos and one sterile neutrino that naturally incorporate maximal oscillations of atmospheric and explain the solar neutrino and LSND results. The models allow either a large or small angle MSW or vacuum oscillation description of the solar neutrino deficit. The models predict ͑i͒ oscillations of either e → or e → s in long-baseline experiments at L/Eӷ1 km/GeV, with amplitude determined by the LSND oscillation amplitude and argument given by the atmospheric ␦m 2 , and ͑ii͒ the equality of the e disappearance probability, the disappearance probability, and the LSND → e appearance probability in short-baseline experiments. ͓S0556-2821͑98͒06921-5͔PACS number͑s͒: 14.60.Pq 4 He yield considerably weaker bounds ͓18͔. Another is that a small asymmetry (n Ϫn )/n ␥ տ7ϫ10 Ϫ5 of flavor neutrinos PHYSICAL REVIEW D, VOLUME 58, 093016