Would a perfectly clean and perfectly smooth champagne glass have no bubbles?
No, the bubbles will still originate inside the liquid, but the process is usually slower. It is called homogeneous nucleation. In nature most nucleation is on surfaces, also called heterogeneous nucleation.
Disclaimer: this is me hypothesizing. Anyone with actual facts at their disposal is encouraged to contradict me.
No, there would be no bubbles formed in the champagne. The reason for this is found in the comment by @honeste_vivere : inside a very small bubble, the pressure (because of surface tension) is very large. In fact, for a bubble of radius $r$, the cross sectional area is $\pi r^2$ and the circumference is $2\pi r$ so the pressure is
$$P = \frac{2\pi r \sigma}{\pi r^2}=\frac{2\sigma}{r}$$
where $\sigma$ is the surface tension (which has units of force per unit length - so multiplying by the circumference gives force, and dividing by area gives pressure).
The problem with this equation is that the pressure becomes "infinite" as the radius becomes very small. And for a very small bubble, this means that the pressure would be so great that $CO_2$ is driven back into solution.
In other words - in the absence of nucleation, there is a minimum bubble size needed for bubbles to be able to grow - and bubbles cannot spontaneously get to that size. When a nucleation site is present, it alters the balance of forces so bubbles can form.
For cold champagne, the surface tension is around 40 - 50 mN/m (source). The pressure inside a closed bottle of champagne is about 6 atm (source)_.
It is reasonable to assume then that if a bubble can get to the size where the excess pressure due to surface tension is less than 5 atm (plus the atmosphere itself makes 6), then it will grow with no problem. What is that critical size?
$$r_{crit} = \frac{2\cdot 50mN/m}{10^5 N/m^2} = 1µm$$
This is considerably smaller than the size of a champagne bubble "at the surface" which is about 0.5 mm (same source) - this makes sense of course, as the bubble, once it starts to form, can keep growing.
But can a 1 µm bubble form spontaneously? I think the answer is "absolutely not". At that size, it contains approximately $1.4\cdot 10^7$ molecules of CO2. I don't believe a statistical fluctuation can achieve such a large bubble - the force pushing the gas back into the liquid would be too large. You need "something" to bring about nucleation.