How can I tell (easily) if my power supply is regulated/switched or unregulated/nonregulated?

Using your voltmeter, just measure the output of the wall-wart without any load. You can generally stick one probe into the middle of the connector, and hold the other against the outside. With a few exceptions, the middle is positive, so use the red lead there, and use the black lead on the outside shell.

Regulated supplies, without any load, should measure very close to the target voltage of 12v. Unregulated supplies will generally have a no-load voltage anywhere from a couple of volts to several volts higher. If they measured 12v without any load, they would have no headroom to take care of the drop due to the load.

There are two basic differfences between regulated and unregulated power supplies: ripple and output voltage variation. Both these things can be measured with a ordinary multimeter.

First, measure the output voltage of the supply with no load with the meter set to DC. Record this as the no-load voltage. Then switch the meter to AC and record that as the no-load ripple.

Second, put a load on the supply and make the same measurements again. The load should not add any of its own noise. A resistor would be good, but old fashioned light bulbs can work too. The load current should be near the maximum the supply is rated for, but not exceeding it. For example, if the supply is "12V 1A", then you want to draw a little less than 1 amp. Something like a 15 Ω resistor would be good, but keep in mind this resistor needs to be big enough to dissipate the power. In this example, it would dissipate about 10 watts. Enough "12V" lightbulbs to add up to 10 W would work too. In any case, record the DC measurement as the loaded voltage and the AC measurement as the loaded ripple.

Well regulated supplies will have little ripple. Anything over 100 mV is suspect. This is the case whether the supply is loaded or not. A unregulated supply could have a volt or few of ripple, especially in the loaded case.

Regulated supplies actively keep their output voltage that same over a wide range of load currents. If the supply maintains the output within a percent over the load range, then it is almost certainly regulated. Anything more than 5% is suspect for a regulated supply.

Of course ultimately it doesn't matter whether the supply is regulated or not, only what its output voltage does as a function of various conditions. If the output voltage stays reasonably steady with little ripple over the whole load range, it should really not matter to you whether that was achieved by regulation, a low impedance transformer, or by a dead fish being waved over it in a mystic ceremony during production.