According to mainstream science, the entire universe is generally agreed to be "flat". Just how do we know that it is flat? How does the data from COBE allow cosmologists to conclude that it is flat and not, say, donut shaped?

I am not sure it is so easy to discover the global shape (topology) of the universe from the CMBR. However, the local geometry is apparent in the CMBR and this looks flat. This is done by carefully looking at the fluctuations in the CMBR.

 

You derive a "power spectrum" from the fluctuations, that is a curve showing how the size of the fluctuations varied with angle. This agrees with the predictions for a flat universe to some reasonable degree of accuracy.

 

Google COBE and Boomerang.

I see. But, what I really want to know is what would the data look like if it was positively curved or negatively curved? And how small does the angle have to be, with respect to the power spectrum, to be able to decide if it is flat, round, etc.

I don't know. You would probably have to find a good graduate text on cosmology. Maybe someone else can recommend something?