This seems to be the cause of endless discussions, many freewheels tossed on the trash heap, and countless hours spent adjusting derailers. One possible simple cause should always be checked first.

Look at the spacer between the smallest sprocket wheel and the next one.

If you see shiny wear marks, as in this photo, the bike may have been provisioned with a chain (often issued as original equipment on many low end bikes) that is really meant for 1- and 3-speed bikes, even if it has 3/32" rollers. It has larger plates around the joint pins that cause it to ride too high in the sprocket valleys, allowing the teeth to catch the rollers and heave the chain forward a link under stress.

Here is a brand new Shimano freewheel on a Giant Cypress ST that was experiencing chain skipping, with its original chain:

You can see from the light between the chain and the sprocket that the chain is not fully seating.

Here are three random chains from my junk box. All have 3/32" rollers and 1/2" pitch, but they all have different inner plate maximum diameters.

The chain plate diameter is not a specification that is published by any chain manufacturers!
Replace that chain with the top one in this picture and your problems just might disappear.

and if you DON'T have the incorrect chain on your bike, and the rear hub is a freeHUB/cassette design, then you may have a bad ratchet in the freehub... the low budget Chinese copy hubs seem to have this problem if ridden hard. I have two here right now with this issue.... I salvaged the axle sets and the rims...
I personally have yet to run across a bad freewheel that didn't have other problems before the ratchet skipped... failing bearings, rain intrusion, etc.

You may need to replace the chain, and freewheel. If you replace chains BEFORE they wear (stretch), your more expensive freewheel/cassette will last for years before requiring replacement!
Cheers

Might you have a frame with very flexible chain and seat stays, allowing the frame to flex when pedaling vigorously.

Absolutely! But don't forget to mention that a failing ratchet will manifest in low gears as well, because the low gear torques the freewheel more than the high gear with any given level of pedal pressure. There is also possibly a failure mode in which worn bearings and hard pedalling cause the entire freewheel to cock, causing the ratchet to lose its grip. This is just a hypothsis on my part, though, but it does possibly explain why the failure is more frequent on the smallest sprocket. Does anyone know if this failure mode exists in the real world?

I've also seen a skip when one tooth was slightly bent. Some pliers fixed the problem. Good call on the wrong chain. I've never seen that before but I'll be remembering that!

The situation you recommend to check first is by far the least likely cause of skipping. In over 20 years of working on bikes there were very, very few instances where a 1/8" chain was installed - maybe 1-2. It is however possible for a similar incompatibility to occur when an old chain designed for a 5 speed freewheel is used on a much higher count rear cassette. In addition the solution you describe will not guarantee that "your problems will disappear."

If the cog cluster is a cassette beginning with an 11 cog, and if the more common lockring is used instead of the special 11 cog lockring, the chain will skate on the lockring instead of fully engaging the cog teeth resulting in chain skipping.

it's a matter of LOAD being applied to the ratchet pawls, not torque reaching the ground.... I've seen the freewheel bearings get nasty loose/worn... and even the hollow bolt holding the ratchet onto the hub come loose... the bad bearings can cause ratchet skip, but the loose freehub bolt did not cause skip.
Riders that constantly insist on "quickly" launching in the small rear gear, or stomping their way up hills, seem to have repeated ratchet failures... it's all about the LOAD being presented to the ratchet, not the torque factor. Torque is action/reaction. Bearing failure comes from rust and/or excessive loading/heat buildup caused by improper adjustment (too darn tight).... mostly rust, in the case of bicycles. Running the wheel hub axle cones too loose will cause high loads to the bearing ball at the top of the cone due to lack of load distribution.... the cone can actually melt and gaul even from being too tight or too loose. Automobile bearing failure can occur from either situation... they are loaded much more heavily than bike bearings, so these failures happen more frequently... first the bearing heats from tightness, then it gets loose from expansion... gauling sets in, adding friction... the grease melts away from the contact surface then too... and you end up walking home, or calling a rescuer on the cel phone these days... technology to the rescue!?!?
and then, there's instantaneous loading during impacts allowed by loose bearing adjustment/slop... rust or dirt intrusion causing excessive heat......
so many failure factors, so little maintenance.... sigh.

Well, you said " meant for 1- and 3-speed bikes," so it was natural to assume 1/8 inch, as that was what was provided on those bikes, whether coaster brake or not. In any case, the one you showed would have more likely supplied on 5 or maybe 6 cog freewheels, not 7 speed.

Thank you,
I was out Craigslist shopping today when the seller mentioned that his girlfriend's bike was skipping on high gear (the only one that was working).

The bike was a really old Schwinn 5-speed in not too bad of shape, just old and worn.

Cranking the chain, I could hear it was going "pink pink pink" when in high gear, but otherwise it shifted well (but the derailleur lever was slipping due to the protective cover being installed wrong). Otherwise, no rubbing on the sprockets. But, I could see the exact wear pattern on the freewheel above, as well as the larger chain plates.

I'll try to get them setup with a new chain and a mini-tuneup shortly.