At what average rate do supernovas happen? By average rate $R$, I mean I want an answer with units $$[R]=\frac{\text{supernovas}}{\text{length}^3 \cdot \text{time}}$$. That is to say, if I consider a huge volume $V$ and a long timeframe $T$, I would expect to see $N = R \cdot V \cdot T$ supernovas occur, where $R$ is the average rate that I'm asking about.
By the way, I'm sure the answer could be incredibly nuanced since the rate probably changes over time / space. But I don't care about those pedantic details. I just want a simple rough average answer based on what we currently observe.
 A: To an order of magnitude:
There is 1 core collapse supernova per century in a galaxy like the Milky Way and about 1 big galaxy like the Milky Way per 100 Mpc$^3$.
This gives your supernova rate as $10^{-4}$ yr$^{-1}$ Mpc$^{-3}$.
The addition of type Ia supernovae might increase the figure by 50%.
These would be figures for the local universe (say, within a billion light years). The number density would be of order 100 times higher 5-10 billion years ago (at redshifts of $z>1$) because the star formation rate and hence core collapse supernova rate in the comoving volume was $\sim 10$ times higher and the comoving volume was $\sim 10$ times smaller.
These numbers check out well against more accurate calculations and measurements in Horiuchi et al. (2009). The plot below, taken from that paper, shows measurements of the core-collapse supernova rate (note this is expressed as a co-moving density - i.e., the expansion of space as $z$ decreases has been removed) over cosmic time (recorded as distance when $z$ is small). The lower plot shows the ratio of type Ia supernovae to core-collapse supernovae. The solid lines show a prediction (and uncertainty) based on the measured star formation rate density as a function of redshift.

A: Depends on what you mean with "huge volume", depends on whether you count all supernovas or just e.g. core-collapse ones, and depends on whether you are wondering today or five billion years ago. But since you are only looking for ballpark numbers... :)
In our Milky Way galaxy, the expected rate of supernova explosions is approximately two per century.
For the rate across the universe, you can find a discussion of the total supernova history of the universe in the context of the neutrinos we may observe from that in the paper by John Beacom. He assembled this plot for his talk at TAUP 2011:

This gives you the ballpark numbers you were looking for. Note that this takes into account that billions of years ago, the star formation and death rate was much higher than today, which is why the numbers are a bit different than what you would expect from a naive integral.
