This business of flipping the plug, using oscilloscopes, or noting hz is all indicative of not understanding Diodes. LED's (Light Emitting Diodes) only work with the current flowing through them in one direction... direct current. Alternating Current (your house current) fluctuates or oscillates 60 times a second.. hence "60 hz". In order for your LEDs to wok on that circuit they need half of those oscillations to be eliminated. They also need the voltage to be greatly reduced.... from 110 down to about 5. LED's also cannot be tested like an incandescent string of lights using an ohmmeter because they have no filament. Testing a bring of LED's would require setting up your multimeter as an ammeter, breaking and testing the light string only (not the other two lines.
Because the string is made to work AS A WHOLE UNIT with AC power, the observation that flipping the plug should alert you that this function has been destroyed. The fuse was there to prevent further damage (fires). The fact that you can replace them should also bring with it the caution that you need to find out what caused the failure in the first place.
A shorted bulb in one of your strings most likely caused a failure in a diode that changes that 110v AC to DC . That event likely caused the fuse to blow. Replacing the fuse or flipping the plug does not address the root problem, and my bet is that you will have to toss the string and buy new.
While you are inspecting your defunct string, break into and have a look at an LED. See that the two electrodes inside are different sizes. The large one is the negative side, the small is the positive. There is space in between... no filament. The polarization of the two poles, and not current flow from one to the otheris what makes that space in between give off light. In fact, if the voltage is too high (AC or DC) that space will be violated and the LED will literally "pop its top". Depending on the element in that matrix, you get different colours. Red is the cheapest light to produce. Clear light is one of the most expensive.