Unique continuation fails on sets where two solutions have agree, so their difference vanishes. So we can look for nonvanishing results on nonzero solutions. Some quantitative control on vanishing of the gradient of a nonzero solution, stronger than but implying an estimate on the Hausdorff dimension of the zero locus of the gradient of any nonzero solution, was proven by

Cheeger, Naber and Valorta, Critical Sets of Elliptic Equations, Communications in Pure and Applied Mathematics, 68 (2015), no. 2, 173–209.

On the other hand, there are counterexamples to various forms of unique continuation, for linear equations (I think some of them elliptic) in

S. Alinhac and M. S. Baouendi, A nonuniqueness result for operators of principal type, Math. Z. 220 (1995), no. 4, 561–568. 85

and for nonlinear in

G. Métivier, Counterexamples to Hölmgren’s uniqueness for analytic nonlinear Cauchy problems, Invent. Math. 112 (1993), no. 1, 217–222.

Unique continuation fails on sets where two solutions agree, so their difference vanishes. So we can look for nonvanishing results on nonzero solutions. Some quantitative control on vanishing of the gradient of a nonzero solution, stronger than but implying an estimate on the Hausdorff dimension of the zero locus of the gradient of any nonzero solution, was proven by

Cheeger, Naber and Valorta, Critical Sets of Elliptic Equations, Communications in Pure and Applied Mathematics, 68 (2015), no. 2, 173–209.

On the other hand, there are counterexamples to various forms of unique continuation, for linear hyperbolic equations in

S. Alinhac and M. S. Baouendi, A nonuniqueness result for operators of principal type, Math. Z. 220 (1995), no. 4, 561–568. 85

and for nonlinear in

G. Métivier, Counterexamples to Hölmgren’s uniqueness for analytic nonlinear Cauchy problems, Invent. Math. 112 (1993), no. 1, 217–222.