It is well known that a smooth vector field on a 2-sphere must vanish twice.
What is the general technique for counting singularities of a smooth map between manifolds? For example, how many singularities must a smooth, surjective map $\phi: V \rightarrow SU(n)$ possess? Here $V$ is a finite dim vector space over $\mathbb{R}$. Can this be established via any analogous method to the vector field on the sphere.
On
Well here's maybe a somewhat enlightening example. Any smooth map $f:M\rightarrow \Bbb R^n$ where $M$ is an $m$-dimensional closed compact manifold has infinitely many singularities (for $n\geq 2$ and $m\geq n$). If $f$ only had finitely many singularities, then $f(M)\subset \Bbb R^n$ would be compact with non-empty interior and hence have infinite topological boundary (for $n\geq 2$). By the inverse function theorem, a topological boundary point must be a critical value.
In fact using a little covering space theory, we can conclude from this that any map from $S^2$ to a closed orientable surface of genus $\geq 1$ also has infinitely many singularities.
Generically, I do not think one should expect there to even exist maps where singularities are isolated, so the case where the target is $\Bbb R$ (as in Morse theory) is quite special.
Edit: Here's a further generalization of the above that should be easy for anyone interested to fill in the details. Let $M$, $N$ be smooth compact manifolds (without boundary) with $\operatorname{dim}M \geq \operatorname{dim}N$. If we have $p(\pi_1(M))$ has infinite index in $\pi_1(N)$ for any homomorphism $p: \pi_1(M) \rightarrow \pi_1(N)$, then any map $f:M \rightarrow N$ has infinitely many singularities.
Concerning your first case. Section 9 of Heinz Hopf's lecture notes on differential geometry is called: "the role of the euler characteristic in the theory vector fields" which generalizes your first statements to surfaces:
Theorem: Summing up the singularity of any vector field on a surface, gives the Euler characteristic.