The engine I'm specifically thinking of is for the C-130J Hercules, it's the AE-2100 D3.
As Bianfable has pointed out, the C-130 Hercules does not have a supercharger or a turbocharger. No turbine-powered aircraft have them. Turbine engines do not need these forced-induction chargers since turbine engines compress the intake air themselves. It would be impractical to design a supercharger or turbocharger for them.
The easiest way to know if a piston-engined aircraft has a supercharger or a turbocharger is to check the aircraft’s POH. If one is not handy, you can look at the aircraft engine itself.
An engine with a supercharger will have an accessory belt (usually wider than the other accessory belts) attached to the crankshaft or flywheel. In older aircraft (WWII, Korean War, etc), the supercharger will be less obvious. These may even be chain or gear-driven. The application of superchargers in this case would be for large, powerful engines.
An engine with a turbocharger will have the exhaust manifolds converge down to one pipe. That pipe will feed into the exhaust turbine of the turbocharger. Although, the exhaust turbine might not be readily visible, the housing for the compressor impeller should be more obvious and may obscure the turbine. What would be more obvious would be the additional manifold piping and ducting.
If neither of the above are accessible, just look at the model name of the aircraft. Most modern small General Aviation aircraft manufacturers will want to advertise that their model is turbocharged. A manifold pressure gauge that goes beyond 30 inches of Mercury may be an indicator as well.
Superchargers and turbochargers can be used on piston engines to increase the manifold pressure above ambient pressure, meaning more air can go into the cylinder during each stroke. This increases the power output of the engine. While superchargers were quite common in high performance aircraft of the past, they are relatively rare today. Most aircraft piston engines today are naturally aspirated or turbocharged:
Back in the mid-1900s, and especially during WWII, superchargers were usually the engine boost of choice. Their simpler operation was one of the main reasons. Another reason they were popular was a lack of metals that could handle the heat produced by turbos. Gas was also very cheap, so it wasn't that big of a deal to burn a little more gas to get the power output you wanted.
High temperature alloys, lighter weights, and better fuel efficiency have all made turbos the best choice for almost all of today's aircraft. And, with automatic waste gates on many models, they're easier (and more foolproof) to operate than ever before.
The Rolls-Royce AE 2100 engines used on the C-130J Hercules are turboprop engines. They don't have any pistons and therefore no super- or turbochargers. A turboprop engine looks something like this:
The air entering the engine goes through a series of compressor stages, which increase the pressure, before going into the combustion chamber. This is more equivalent to the compression stroke of the piston engine, but like a supercharger this compressor is powered by the engine itself (it is on the same shaft as the turbine). Every turbine engine has such a compressor, but it is never called a supercharger.
To answer your title question, how do you know if an aircraft engine has a supercharger? For a piston engine, it will be specifically mentioned, since it is relatively special. But for a turbine engine: never.
As in immediate recognition, for spotters for example, the easiest way to tell the difference is by the sound they make, more specifically on take-off. Turbines have a fairly high singing pitch, due to their rpm, whereas piston engines, making a generally much lower pitched brawling sound, only have such a high pitch added to their sound profile if they are either turbocharged or supercharged. Not a very technical answer, but if you just want to be able to tell the difference, that's a way to do it.