We exploit here the dramatic field enhancement caused by energy squeezing and tunneling i.e., "supercoupling" in metamaterial-inspired ultranarrow waveguide channels with near-zero effective permittivity in order to sense small permittivity variations in a tiny object. The supercoupling effect is accurately modeled analytically and closed-form expressions are derived to describe the presence of defects or permittivity perturbations along the channel. Applications for tailoring its pass-band frequency and for high-Q sensing are proposed at microwave frequencies.