I was watching some prototype videos of European trains and noticed the following: Engines running with the forward pantograph raised: Engines running with the rear pantograph raised and engines running with both front and rear pantographs raised. Is there a rule as to which pantograph to use? The only thing I know is engines running with the front pantograph is not good practice because if the pantograph should get snagged and ripped off, it could also rip off the rear pantograph and you would be left with no pantograph.

Yes, there are many such rules, but they differ from railway to railway and are also not identical for all types of trains and their loads. And some of the rules also contradict each other.

For instance, there is a rule that when a cargo train carries factory-new automobiles, then the rear pantograph of the locomotive should not be raised because flying "rub off" from the pantograph's wiper could damage the load on the first wagon.

Furthermore, in case of a double-heading (two pulling locomotives), the pantographs should be raised as far apart as possible (example: front pantograph on the leading locomotive, rear pantograph on the second one) to minimize bouncing and thus interruption of contact.

A third practice is often applied to cargo trains carrying inflammable loads (petrol, kerosene) where also the front pantograph should be used due to the danger of flying sparks from the pantograph.

And of course, the general practice regarding the preference of the rear pantograph has already been mentioned in your initial posting.

Edit: I forgot to mention one situation when both pantographs are raised: In very cold winter days, the first one serves as an "ice scraper", the second one transmits the power. However, in such situations, restrictions with regard to the speed of the train mostly apply.

I think Unholz gave a very good answer, but I think there is one detail more. (I might be wrong)
Some older e-loks should run with both pantos up always, if the ski on the panto is "single". -As in older times.
If the lok has been equipped with a more modern pantographs with doble slider, then only one should be raised.

A totally different issue; I sometimes see people running their restaurant cars with the panto up. That is a NO-NO, as these are only used when the waggon is without a connection to a lok.

Anther interesting pantograph use I’ve seen at some border stations Jeumont on the French/ Belgium frontier, we’re the use of switchable catenary from the SNCF system to the Belgium system is used. Despite being switchable, the trains use to roll to a stop with there pantograph’s down? Nowadays the multi voltage locos run straight through with the pantograph’s for each system being changed on the move. This use to happen with the SNCF 40100 and SNCB equivalent, but all freight trains use to do this.