Toilet paper dilemma
I'll propose a theory, and I'll describe an experiment I did to test it. Both suggest that the "over" configuration is better, at least if the goal is to make the squares easier to rip off with one hand without making the roll spin out of control.
Please don't use this post as ammunition to defend a preference.
As rightly emphasized in several good comments (moved to chat), the specific issue addressed in this answer is not the only issue or even the most important one, and it assumes specific conditions that probably don't represent most real-world situations. I only posted it because I thought the experimental results were amusing. After seeing how much attention this answer has been getting, I decided to add this terms-of-use clause in case the answer's lighthearted intent wasn't already clear from the writing style. :)
Toilet paper physics: theory
The diameter of the cylindrical hole in the toilet paper is larger than the diameter of the axle on which it rotates, as illustrated in the pictures below. So when the toilet paper is at rest, only the top of the hole is in contact with the axle. The bottom and sides of the hole are not in contact with the axle.
That matters because when the toilet paper is in the "over" configuration, you can tear off a square by yanking it straight down. Since the top of the hole is already in contact with the axle, the roll's resistance to the yank is immediate, and the square rips off easily.
In contrast, when the toilet paper is in the "under" position, the loose square is hanging close to the wall. In that case, yanking straight down is hard to do without scraping your hand against the wall. We can get around that by yanking at an angle (or horizontally) instead, but then the roll's resistance is not so immediate because the opposite side of the hole is not in contact with the axle when the yank begins. The roll must move laterally to bring the opposite side of the hole into contact with the axle. That delays the rip and increases the tendency for the roll to spin, which tends to wind the paper up around the roll in addition to (or instead of!) ripping it off.
Toilet paper physics: experiment
I tested this by removing the axle from the fixture and holding it in one hand while using the other hand to yank on the loose end of the paper. I tried all combinations of the following options:
Hold the axle with the left hand and yank with the right, or hold the axle with the right hand and yank with the left.
Hold the roll in the "under" configuration, with the paper hanging on the opposite side from the hand I used to yank the square, or hold the roll in the "over" configuration, with the paper hanging on the same side as the hand I used to yank the square.
Yank straight down, or yank at a near-horizontal angle. In both cases, the angled yank was toward the side with the hand I used to yank (toward the left when yanking with my left hand, and toward the right when yanking with my right hand). In the "under" configuration, this meant yanking toward the opposite side from the side on which the paper was hanging. In the "over" configuration, this meant yanking toward the same side from the side on which the paper was hanging.
Here are the results:
Didn't matter which hand I used to hold the axle or yank the paper, even though I'm not ambidextrous.
In both the "over" and "under" configurations, yanking straight down almost always ripped off the square without spinning the roll significantly at all. (I tried a few times in each configuration, and there were only a couple of exceptions overall.)
In the "over" configuration, yanking at an angle still almost always ripped of the square without spinning the roll significantly, and the number or exceptions was only slightly greater than when yanking straight down.
In the "under" configuration, yanking at an angle almost always made the roll spin at least one full revolution, unwinding much more paper than intended, and it often failed to separate the square at all.
Here's a graphic summary:
These results seem mostly consistent with the theory described above, and they suggest a refinement. When yanking at an angle in the "over" configuration, the straight segment of the toilet paper is tangent to the upper part of the roll before the yank, close to the point where the hole is resting on the axle. When yanking at an angle in the "under" configuration, the straight segment of the toilet paper is tangent to the lower part of the roll before the yank, far from the point where the hole is resting on the axle. As a result of the longer lever-arm, yanking at an angle in the "under" configuration applies a larger torque (hence a stronger tendency to spin the roll) compared to yanking at an angle in the "over" configuration, as observed in the experiment.
Other things to try
What if the roll's hole fits snugly onto the axle, so the hole is in contact with the axle around its whole circumference? The theory predicts that this should eliminate the differences between the "over" and "under" configurations. I didn't test this.
I always tried to yank the square orthogonal to the perforated line. I didn't try oblique yanks, to make the rip propagate more gradually from one side of the perforated line to the other.
The proposed theory doesn't explain the correlation with right- and left-handedness cited in the question.
This answer only considered one goal: ease of removing a square with one hand without causing the roll to spin out of control. Different goals might be better served by different configurations.
Disclosure of conflict of interest
I've always preferred the "over" configuration. I tried to conduct the tests fairly, because I really was curious, but I suppose my prior preference could have had some subconscious influence on how I did the tests.
I don't think I need to explain the physics of tearing something but there are two givens:
closer you are (the pivot) to the tear point the easier - so you want your hand as close as possible to the roll.
the faster your tear the easier the toilet paper will rip. So the faster you do it the better.
Given that if you have the toilet paper under (for righties or lefties) your hand will be further from the pivot and you will not be able to tear as fast.
The under presents someone with having no flexibility in allowing a one hand tear. This is surely where magazines get dropped on the floor and cell phones broken.