Well the most important thing is to actually think about everything you're about to do when connecting a circuit.
Check if the oscilloscope probe's ground pin is shorted to the power supply's ground pin (in most cases it will be). Check if output pins of the power supplies are connected to ground pin. They might be, but good quality supplies will have separate ground pin available on front. Also check if the PSU case if connected to ground (it probably will be) and if it is, take care not to have a positive wire touch it.
Every time you connect the scope to something powered by the power supply, think what's going to happen. I've seen cases where people shorted-out their power supplies using oscilloscope probe ground pin and couldn't realize why that happened.
Next, check how the power supplies are going to react in overcurrent situation. Is it going to shut down or will it drop voltage or something else? In general, familiarize yourself with equipment you'll be using.
Do keep in mind that a multimeter in current measurement mode is basically a short-circuit and in voltage mode is basically open circuit. Take care how you connect it! Do read the manual and try to understand what happens in other modes, if there are any. Take note of maximum voltage the multimeter can take with each of them.
For the end, once again, read the equipment manuals and try to plan out each situation in which you can find yourself in as far as the equipment is concerned and think about what will happen if you make a short somewhere.
ABOUT THE COMMENT:
The problem has to do with "floating" and "ground-referenced" power supplies. When a power supply is said to be floating, it means that you can't make a current loop which goes back to ground. An example of this is a battery-operated device. Current goes from one terminal of the battery to another and if you connect one side of the battery to the ground, no current will pass through the wire, because there isn't a closed loop for current to go through. Take a look at this simulation where the ground is connected to the positive output of the battery and no current goes through it.
Same thing happens when you have a transformer separating the mains side of a power supply from the low voltage side of the power supply. All current going out of the secondary side of the transformer needs to go back into the secondary side of the transformer and if you touch a wire along the path, no current should go through you since there isn't a loop for it to go through. Take a look at this simulation. Here too we have ground on the secondary side of the power supply and co current goes through it.
Now to get back to measurement instruments. A hand-held multimeter is often battery powered and is therefore isolated from the circuit it is measuring. This allows you to for example connect the negative probe of the multimeter to the positive pin of the power supply and positive probe of the multimeter to the negative pin of the power supply and the measurement will work, but you'll get negative voltage.
On the other hand, most oscilloscopes are connected to mains power and the ground pin of the probe will usually be connected to the ground pin of the oscilloscope's power supply. On some bench-top power supplies, the negative side of the secondary is connected (or can be connected) to ground. If you for some reason connect the ground pin of the scope probe to the positive pin of the power supply, you can create a short. It will look something like this.