SIM800L restart automatically
You can't really read the voltage during the current spikes, the multimeter is not quick enough.

GSM modules induce a large current spike during transmission and even larger during network registration (~2A) for ~500uS. The regulator might not to be able to provide it that quickly. Put a ceramic capacitor (1uF) in parallel with a bigger electrolytic one, like 1000uF on the output. Don't forget to put them as close as you can to the module's pins, especially the smaller one. Good jumper wires will help too.
Not sure why it goes up to 5v though, you might want to try to use a different regulator/power supply. If you're using a 3.7V lithium battery, you can just feed the module directly without the regulator.