lacking further clarification of the question, heres a try/sketch of an answer. matrix rigidity has deep connections to fundamental questions in TCS/complexity theory including circuit lower bounds, and thereby complexity class separations, and coding theory as well as other areas.  is a nice slide survey.
the terms "low" and "high" in reference to rigidty of matrices are used informally and not in a precisely defined technical sense. [although Friedman did define "strong" rigidity. ] random matrices are known to have high rigidity but basically, its ~a 3.5 decades old open problem in this area to explicitly construct any matrix with "significantly high" rigidity.
the question doesnt further define/clarify the subjective terms "nontrivial" or "nonobvious" & will take some freedom there.
in this area there is a line of research looking at the rigidity of Hadamard matrices which have misc uses/applications in coding theory & elsewhere.
it seems fair to say a provably high rigidity result would surpass the threshhold of leading at least to "new nontrivial corollaries in complexity theory" but the best known bounds on Hadamard matrices do not suffice. but neither does this conclusively prove they have limited "low" rigidity. its basically the same story with Vandermonde matrices [also applications in coding theory] considered by Lokam.
so to summarize about all that can be said is that "weak lower rigidity bounds" have been proven on some matrices including Hadamard/Vandermonde matrices.
there also do not appear to be any published numerical experiments, estimates, or algorithms in the area.
 Boolean Function Complexity by Stasys Jukna, 2011, sec 12.8 "rigid matrices require large circuits"
 On matrix rigidity and locally self-correctable codes Zeev Dvir
 Improved lower bounds on the ridigity of Hadamard matrices Kashin/Razborov
 On the Rigidity of Vandermonde Matrices Lokam
 Mahdi Cheraghchi matrix rigidity talk
 J. Friedman. A note on matrix rigidity. Combinatorica, 13(2);235-239, 1993