It is not difficult to tile the plane with incongruent triangles. One could tile with equilateral triangles, and then partition each equilateral into three triangles, displacing their common centerpoint so that no two triangles are congruent (left below).
Q1. Is it possible to tile the plane with isosceles triangles, no two of which are congruent?
It is easy to tile the plane with congruent isosceles triangles, as illustrated right above. But I don't see how to achieve a tiling with incongruent isosceles triangles.
Perhaps it is easier to answer this question:
Q2. Is it possible to tile the plane with equilateral triangles, no two of which are congruent?
Added 14Feb2020: Q2 has been answered (negatively) in two papers (independently). These results were presaged (by @Wojowu) below.
(1) Pach, János, and Gábor Tardos. "Tiling the plane with equilateral triangles." arXiv:1805.08840 abstract (2018).
Corollary 4. There is no tiling of the plane with pairwise noncongruent equilateral triangles whose side lengths are bounded from below by a positive constant.
(2) Richter, Christian, and Melchior Wirth. "Tilings of convex sets by mutually incongruent equilateral triangles contain arbitrarily small tiles." Discrete & Computational Geometry 63, no. 1 (2020): 169-181. Springer link.
Question Q1 was inspired by (but not addressed in) this paper:
Malkevitch, J. "Convex isosceles triangle polyhedra." Geombinatorics 10 (2001): 122-132.