Many recently synthesized materials feature aligned arrays or bundles of carbon nanotubes (CNTs) whose mechanical properties are partially determined by the van der Waals interactions between adjacent tubes. Of particular interest in this paper are instances where the resulting interaction between a pair of CNTs often produces a forklike structure. The mechanical properties of this structure are noticeably different from those for isolated individual CNTs. In particular, while one anticipates buckling phenomena in the forked structure, an adhesion instability may also be present. New criteria for buckling and adhesion instabilities in forklike structures are presented in this paper. The criteria are illuminated with a bifurcation analyses of the response of the forklike structure to applied compressive and shear loadings.