SimCom64

There are several different SIM800 modules out there. Make sure that the one you choose has a level shifter for UART pins - otherwise you need to add level shifter your self - since normally SIM800 RX/TX pins cannot handle 5V.

Do not power the module from user port. Instead a separate 5V power supply should be used - which is happy to handle ~2A
... The module doesn't actually take such a huge average current but there are high peaks during cellular transactions.

These modules usually need a Li-Ion battery to be connected. If you try without a battery the module starts but keeps resetting when cellular
peaks occur since the battery is not there stabilizing the current peaks.
Downside of using a battery is that the module is left on if you switch off the power - until the Li-Ion battery runs out.
You may go around this problem by using a large enough -capacitor instead of the Li-Ion battery. A 4700uF electrolyte capacitor seems to stabilize the current peaks nicely and the module shuts down after 2-3 seconds when external power is shut down.

Not necessarily a bad idea to add some additional components to limit the inrush current (high current when empty capacitor gets charged). The module I'm using hasn't suffered even though used quite a lot with the 4700uF capacitor directly connected to battery pins, but cannot promise every module would be fine with this.

Connecting module:

A few pictures of the module with the connectors assembled as well as from the app: