How fast do you have to spin an egg to have it standing?

I think one suited approach to your problem is to look at the stability of the movement with respect to the energy.

Energy stability analysis

The rotation energy is given by $$E_{rot} = frac{1}{2}Theta{(mathbf{omega},mathbf{omega})}$$

where $Theta$ is the (tensorial) moment of inertia with respect to the (vectorial) rotation $mathbf{omega}$.

The mass of a prolate, homogeneous ellipsoid is given by $M = frac{4}{3}rho r_a^2r_b$ where $rho$ is the mass density and $r_b$ is assumed to be the axis longer (or, say different) than $r_b$.

Then, we can write down the principal moments of inertia for this ellipsoid as $$Theta_b = frac{M}{5}cdot 2 r_a^2$$ and $$Theta_a = frac{M}{5}cdot (r_a^2 + r_b^2)$$

If we now chose $theta$ to be the angle between $mathbf{omega}$ and the $z$-axis, we find the rotation energy as $$E_{rot} = frac{M}{10}left( 2r_a^2 cos^2theta omega^2 + (r_a^2 + r_b^2) sin^2theta omega^2 right)$$

with $omega$ now being the scalar value of rotation (I could not make it bold).

Now, we have to turn to the potential energy in the gravitational field. I suppose, this should take the form $$E_{pot} = Mgleft( r_b costheta + r_a sintheta right)$$

If you now analyze the energy $$E = E_{rot}+ E_{pot}$$

with respect to $theta$ and the parameters $r_a$ and $r_b$, you will find out that $partial_theta E$ will vanish at $theta = 0$ but it will be a maximum (remember, $thetageq 0$) assuming $r_b > r_a$.

That means, that the movement will be unstable assuming a rotating prolate body. Or, in other words, you cannot find some $omega_s$ to have the egg at resting rotation in this model.

Other approaches

One further ansatz to calculate a stable rotating egg, one could assume that the rotating egg is not homogeneous. A simple model could be to assume that a small ring of mass is attached to the "smaller belt" leading to $$Theta_b = frac{M}{5}cdot 2 r_a^2 + frac{M_{ring}}{2}r_a^2quad, qquad Theta_a = frac{M}{5}cdot (r_a^2 + r_b^2) + frac{M_{ring}}{2cdot 2pi r_a}r_a^2$$

Now, can you show, when $E$ has a minimum in $theta = 0$ leading to a stability condition of the form $F(M_{ring}, r_a/r_b; omega_s) = 0$?

I think that if you could spin the egg perfectly on its edge, then it wouldn't stand up. It stands because of a wobble as it spins. This wobble changes the stability of the egg's rotation, causing it to stabilize itself. If you do this enough times, you'll notice that the egg will sometimes stand on its sharp edge. Also, this can be done with a football, so the matter of "one side of the egg is bigger" isn't relevant.