Can it be argued that most choices of $f:M^{\rm2} \mapsto \mathbf{\mathbb{R}^{\mathrm{3}}}$ will be mostly injective? It seems intuitively that if $f:M^{\rm2} \looparrowright \mathbf{\mathbb{R}^{\mathrm{3}}}$ for some choice of function(s), then most functions that are in some sense “well behaved” should produce a surface in $\mathbf{\mathbb{R}^{\mathrm{3}}}$ that is mostly injective.
Is this intuition sensible?
 A: This seems like a soft question, and my initial response is that you're intuition is correct.
First (please forgive me if this is all familiar): Surfaces (even different patches of the same surface) intersect generically in curves. These are called transverse intersections, where the tangent spaces of the surface patches together span the tangent space of the ambient $\mathbb{R}^3$.
A nice way to keep this straight in your head is to think in terms of codimension instead of dimension. If $L_1$ and $L_2$ are submanifolds of dimensions $\ell_1$ and $\ell_2$, respectively, immersed in $M$ of dimension $m$. Say that $L_1$ and $L_2$ intersect transversely to form submanifold $N$ of dimension $n$. Then the codimensions add:
$$
(m - n) = (m - \ell_1) + (m - \ell_2)
\qquad\Longleftrightarrow\qquad
n = \ell_1 + \ell_2 - m
$$
So, for example in $\mathbb{R}^3$ $(m = 3)$ with surfaces $(\ell_1 = \ell_2 = 2)$. Transverse intersection will be in dimension
$$
n = 2 + 2 - 3 = 1,
$$
i.e., curves.
Now questions of what happens "generically" or "usually" can sometimes be made precise by using measure theory. See, e.g., Sard's Theorem which shows that the set of critical values of a smooth function between manifolds is a null set (measure zero). So, if you're picking values in the image of the function, you have probability $0$ of picking a critical value.
Perhaps you can frame your question in terms of self-intersections of a generic immersion in a mapping space (space of all such immersions)?
