Claim: The group of orientation-preserving diffeomorphisms of the plane modulo isotopy is trivial (that is, any such diffeomorphism is isotopic to the identity).
Proof: If we have such diffeomorphism $f$, we can assume $f(0) = 0$. The isotopy $F(x,t) = f(tx)/t$ shows $f$ is isotopic to a linear isomorphism, namely $df_0$. A linear isomorphism can be isotoped to the identity (by fixing a frame in the domain, rescaling and rotating their images, one at a time, and extending by linearity at each step). This is done in Milnor's "Topology from the Differentiable Viewpoint".
Question: I have not been able to find a reference or prove that the group of orientation-preserving homeomorphisms of the plane modulo isotopy is trivial as well. This is also called the Mapping Class Group (hence the title). Does anyone know how to approach this question?
I believe there's a question involving the closed disk, but this is slightly different, since I don't have compactness. Mapping Class Group of Disc Proof
Edit: In the paper https://math.berkeley.edu/~kirby/papers/Kirby%20-%20Stable%20homeomorphisms%20and%20the%20annulus%20conjecture%20-%20MR0242165.pdf ,first page, (Added December 1, 1968), he claims that the stable homeomorphism conjecture is a classical result for $n\leq 3$ (there's a typo), but he doesn't give a reference.