How much of an $n$-dimensional manifold can we embed into $\mathbb{R}^n$?

This question was essentially answered here, namely, that the cut-locus has measure zero (see the references provided in the link). I assume that your manifold $M$ is smooth, otherwise, I am not sure what notion of measure zero you would be using. I will also assume that $M$ is connected. (If not, apply this argument to each connected component.) Then, put a complete Riemannian metric on $M$, consider the cut-locus $C(p)$ of a point $p\in M$ and the open subset $U(p)=M\setminus C(p)$. The exponential map $\exp_p: T_pM\to M$ restricts to a diffeomorphism $V(p)\to U(p)$, where $V(p)\subset T_pM$ is a certain open subset (diffeomorphic to the open $n$-ball, where $n$ is the dimension of $M$).

One can ask a similar question in the context of connected $n$-dimensional topological manifolds $M$. Instead of removing a measure zero set, one can remove a closed nowhere dense subset. Then in all dimensions but 4 it follows from the results in the book by Kirby and Siebenmann that $M$ contains an open subset $U$ homeomorphic to $R^n$. (Maybe it was also known prior to their work, I am not sure.) I do not know what to say about dimension 4.

One last thing, Sullivan proved that every topological manifold $M$ of dimension $\ne 4$ admits a unique Lipschitz structure, i.e., an atlas where transition maps are locally Lipshitz. This implies that topological $n$-manifolds ($n\ne 4$) have a well-defined notion of measure zero sets (very indirect though). However, I do not know if the residual set in Kirby-Sibenmann will have measure zero in this sense, this goes well beyond my understanding of the work of their work and the one by Sullivan.

Edit. A better reference for topological manifolds (instead of Kirby-Siebenmann) is in my answer here, to papers by Brown (compact case) and Berlanga (noncompact case).