Subject to compression, elastic materials may undergo bifurcation of various kinds. A homogeneous material forms creases, whereas a bilayer consisting of a stiff film and a compliant substrate forms wrinkles. Here, we show several new types of bifurcation behavior for bilayers consisting of films and substrates of comparable elastic moduli. Depending on the ratios of moduli and thicknesses of the two materials, the critical strain for the onset of creases can be either smaller or larger than that for the onset of wrinkles. When the critical strain for the onset of creases is lower than that of wrinkles, creases can be subcritical or supercritical. When the critical strain for the onset of wrinkles is lower than that of creases, wrinkles can further channel to creases at a strain much lower than the critical strain for the onset of creases in a homogeneous material. Experiments, conducted with bilayer polydimethylsiloxane (PDMS) structures subject to compressive loading, show that the different types of bifurcation behavior agree with the theoretical predictions.