Supermoon is defined as:
new or full moon which occurs with the Moon is within 90% of its closest approach to Earth in a given orbit
I'm going to focus on the full moon case because the New moon case is pretty boring since we can't really see it.
Does the Supermoon have a Supereffect on us? (earthsky.org)
Biggest Supermoon for 70 years (meteorwatch.org)
What happens is that as the Moon goes around the Earth, and the Earth goes around the Sun they are not always in the same relative position to each other in the Sun/Earth/Moon system so it goes through cycles. One effect of this is that the exact Apogee (furthest distance of the Moon to the Earth) and the Perigee (closest distance of the Moon to the Earth) distances vary as the extra gravity of the Sun pulls the moon a little further away on one side and keeps it closer to the Earth on the other side. In addition, the moon isn't always in the same position in its own orbit each year because the lunar and Earth cycles do not perfectly sync up so the effect of this pull also varies with those cycles.
With a great deal of exaggeration and utter lack of scale this diagram attempts to illuminate this pull on the lunar orbit (apogee is also not always towards the Sun, but it's pulled that direction when it is).
|Figure 1. the Sun pulls on the Moon and affects the orbit in measurable ways|
Some of the other cycles affecting the lunar orbit are shown here:
|Figure 2. By Geologician, Homunculus 2 - from English Wikipedia, CC BY 3.0, Link|
So what can we expect to see from the Earth?
As you may recall, the apparent angular size of an object g units wide and d units distant is given by the formula:
Using this we can compare the angular size of the moon at apogee and perigee - using a lunar radius of 1738.1 km and comparing the angular size at various distances. You may note that I just compare the lunar radius - since we're doing a ratio doubling it doesn't change the ratio.
Comparing apogee vs perigee we find that supermoons are about 14% larger than micromoons.
And looking at the variations between supermoons during the 21st century, data from astropixels.com, I found it varied by just 3.2% between largest and 'smallest' of the supermoons (based on perigee at 356429 km and 367980 km respectively).
Naked-eye observers are hard pressed to see the 14% difference from micromoon to supermoon (until you put them side-by-side or in scale with something else) but the 3.2% variation between supermoons would be even harder to notice, but it's sufficient enough to measure carefully.
These variations are measurable and are some of the many observable phenomena that aren't explained by any Flat Earth model which has to give ad hoc explanations for everything whereas the scientific model has extensive predictive power.