How many polynomial Morse functions on the sphere? Let $f$ be a homogeneous polynomial of degree $d$ in $n$ variables. Restricted to the unit sphere $S^{n-1}$, it might or might not be a Morse function.
If $f$ is a Morse function of degree $1$, you get everyone's favorite Morse function on the sphere, with two critical points, corresponding to the minimal CW decomposition.
If $f$ is a Morse function of degree $2$, you get everyone's second-favorite Morse function, corresponding to the minimal regular CW decomposition with two points, two lines, two faces, and so on.
If $f$ has degree $d\geq 3$, there are a lot more possibilities. How many?
The space of degree $d$ homogeneous polynomials in $n$ variables can be identified with $\mathbb R^N$ where $N={\left( \begin{array}{c}n+d-1 \\ n-1\end{array}\right)}$. The ones that are Morse functions form an open subset.

About how many connected components does this open subset have for large $d$ and/or $n$?

 A: I've been trying to answer this question for several years and it turned out to be really  hard, even for the $2$-sphere.   Below I will discuss this case.
First of all  one should  ask   what  is the number $m(k)$ of topological types  of (stable) Morse functions on $S^2$  with  precisely $k$ saddle points. (such a function has $2k+2$ critical points.)  I showed that the generating series
$$ x(t) := \sum_{k\geq 0} \frac{m(k)}{(2k+1)!} t^{2k+1}, $$
is the inverse  of an elliptic integral; see  this paper.  More precisely $x(t)$ is the inverse of the function
$$ x\mapsto t(x)=\int_0^x \frac{ds}{\sqrt{s^4/4-s^2-2sx+1}} ds. $$
This fact   leads to a positive answer to a question of V.I. Arnold   who conjectured that
$$\log m(k)\sim 2k\log k $$
as $k\to \infty $. I refer you to  this paper for details.   This shows that $m(k)$ grows rather fast as $k\to \infty$.
Any   polynomial $P$ of degree $d$ in $\newcommand{\bR}{\mathbb{R}}$ on $\bR^n$   can  be uniquely decomposed  as a sum
$$ P= \sum_{0\leq j+2k\leq d} r^{2k} H_{j}, \;\; r^2= (x_1^2+\cdots +x_n^2), $$
where $H_{j}$ is a darmonic polynomial of degree  $j$.      On $\bR^3$ the space of degree $d$ hormonic polynomials has dimension  $2d+1$.   If we denote by $U_d$ the subspace of $C^\infty(S^2)$ consisting of the restrictions to $S^2$   of the  polynomials of degree $\leq d$  we deduce that
$$\dim U_d=\sum_{0\leq k\leq d} (2k+1)=(d+1)^2. $$
Denote by $K_d$ the expected number of critical points of a random function in $U_d$. I showed  that
$$ K_d\sim C\dim U_d\sim Cd^2 $$
as $d\to \infty$ where $C$  is a certain explicit constant; see this paper and this paper.
It turns out that the   number of critical points of a random function in $U_d$ is     highly concentrated around its mean $K_d$, i.e., the probability that the number of critical points of a random function  in $U_d$ is far from the mean  $K_d$ is extremely small as $d\to\infty$.  In more precise technical terms, the variance of the  (random) number of critical points of a (random) function in $U_d$  has the same size  as $K_d$, which makes the standard deviation  of size $\sqrt{K_d}$, much, much smaller than $K_d$ for $d$ large.
I personally believe, based on some empirical evidence,   that the mean  is close to the  maximum number of critical points    in the sense  that if  we denote by $\mu_d$ the maximum number of critical points of a  Morse function in $U_d$, then $\mu_d \sim C'' d^2$ as $d\to\infty$.
My guess is that the number of topological types of functions in $U_d$  as $d\to \infty$ is roughly
$$ \sum_{k=1}^{K_d/2} m(k), $$
where I recall that $m(k)$ denotes the number of topological types  of Morse functions with $k$ saddle points, i.e., $2k+2$ critical points.
