Lightsaber emitters are typically round:
Also, if these lightsabers were anything other than round you would expect them to appear differently depending on the relative angle between the viewer and the lightsaber. However, you can usually see that the dimensions of the lightsaber beam are consistent from every angle, which implies they are circular.
The very first time we saw a lightsaber in Star Wars, Luke holds it at a variety of angles:
(EDIT: though here is an example like OP mentioned where there are brief moments where it appears flat, oh well.)
And as OP and Jack B Nimble mentioned the Darksaber appears to have a flat blade.
The Darksaber's hilt is noticeably different from the typical round emitters:
And when it is moved around by characters wielding you can see that from some angles it appears wider and others more narrow, which is what you would expect for a flat blade. This is demonstrated nicely in this gif:
The Darksaber is a unique kind of lightsaber and thus this difference reinforces the idea that lightsabers are typically round.
Here are a couple more shots that seem to show a round blade:
And finally, for what it's worth, note that the lightsaber props have sometimes been wooden dowels, which are round. If they were going for something other than round, they'd probably have used something else.
EDIT: another thought I had was that really you could only know the answer to this if you were looking at a canon-confirmed lightsaber in 3D. Fortunately, there's a canon VR game "Vader Immortal" that features lightsabers. I haven't played it but I suspect the lightsabers look pretty round in VR. Can't say definitively from this gameplay video but seems round to me. 😜
As pointed out in the comments by @aleppke, you can look at the holes lightsabers make. An example I found of this is in Clone Wars (the hole there looks pretty circular to me):