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Embedding type I seesaw in GUTs, left-right gauge theories, or even in extensions of the SM requires large right-handed neutrino masses making the neutrino mass generation mechanism inaccessible for direct experimental tests. This has been circumvented by introducing additional textures or high degree of fine-tuning in the Dirac neutrino or right-handed neutrino mass matrices. In this work we review another new mechanism that renders type I seesaw vanishing but other seesaw mechanisms dominant. Such mechanisms include extended seesaw, type II, linear, or double seesaw. The linear seesaw, double seesaw, and extended seesaw are directly verifiable at TeV scale. New observable predictions for lepton flavor and lepton number violations by ongoing searches are noted. Type II embedding in SO(10) also predicts these phenomena in addition to new mechanism for leptogenesis and displaced vertices mediated by gauge singlet fermions.