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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?

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chroma
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    Have a look at Stable Mappings and Their Singularities by Golubitsky and Guillemin, in particular chapters Vi and VII. That should give you a fairly complete picture of "generic" maps from surfaces to 3-manifolds (and more). You'll also see that if you want a self-transverse immersion, you'll have get rid of so called cross-caps (or Whitney umbrellas). – Daniel Kruse Feb 18 '21 at 22:10
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    How do you want to make sense of "likelihood" or "well behaved"? For stable maps (which are open and dense in the relevant mapping space) the critical point and double point loci will always have measure zero in the surface, even in the presence of cross caps. Stable maps without cross caps are immersion with normal crossings. – Daniel Kruse Feb 18 '21 at 23:51
  • Sorry, but this is still too vague. It seems to me that your reasoning is something like this: "If an interval is large enough to contain an integer, than most numbers in that Interval should be integers." that's obviously false. Similarly, immersions are very special points in the mapping space. – Daniel Kruse Feb 19 '21 at 08:21

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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)?

Sammy Black
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    Note that it is not true that a map $\Bbb{RP}^2 \to \Bbb R^3$ is generically an immersion, let alone a self-transverse immersion. For example, a cross cap - or Whitney umbrella - is stable under small perturbations, it doesn't resolve. However it is true that the curve of singularities is at worst $1$-dimensional. If I recall correctly Boy's example comes from cleverly cancelling a pair of cross caps. Immersions of $\Bbb{RP}^2$ in $\Bbb R^3$ are kind of surprisingly special; for example, they always have triple points. – Balarka Sen Feb 18 '21 at 11:03