and all of the above, but as we are now two steps farther in on the thread structure I don't remember the order in which you asked the questions
There is no easy way to ''test'' which item is failed without taking it apart. Generally speaking, power supplies are frequent culprits, so you can often get ahead of the curve by getting a new one from your local shop/ BustBuy/ ChumpUSA/ or whatever. If, a few steps farther down the troubleshooting path, you find you don't need it because the problem is found to be something else, you can probably return it for your money back. But having it on hand when you open up your system will save you running out in the middle of the job to get the part that we could have told you in advance is 90% likely to be needed. So for the rest of my reply let's just assume you now have a new one.
The power switch on the front of the case is just a momentary contact switch. The motherboard always has a little bit of power, and pushing the contact switch simply completes a circuit between two little prongs on the motherboard that signals the power supply to start supplying full juice to the system. When you open the case and trace the wires from the backside of the on/off switch down to the motherboard, you can pull the those two wires off the motherboard and touch the two prongs with a small screw driver (be careful poking around - touch only the two prongs that were underneath the wires from the switch). The system should come to life. If so, then the switch is bad, it's not making the momentary contact to complete the circuit as it should. Replacing it will be a chore as there are tons of styles and sizes to choose from. But I doubt the switch went bad, so reconnect the wires to the prongs on the motherboard.
Step 2 - I suspect the system will not come on even from the screw driver ''jumper.'' That means either the motherboard is bad or the power supply is bad. If the motherboard is gone, that is really bad news, but less likely than the power supply. As noted at the beginning, power supplies crap out all the time. Since it is both easier and cheaper to replace the power supply than pull it and send it out for testing, that would be the next step. The supply itself is held into its rack with 4 screws on the back. The tangle of power leads will need to each be disconnected at its termination (one or two will be directly on the motherboard, several will go to the backends of drives, fans, and other peripherals). The connectors are sometimes tricky to wiggle apart, but they are all standard size/types so for everything you take out for the old supply there will be one just like it on the new one. You just mate them all back in the manner you took them apart. When you get all the connectors back together, plug it in a try again. A dollar to a donut, it will start up. Satisfy yourself that it works, and then turn it off. While you have the case open, clean out the dust bunnies and the heat sinks and fans. That will improve cooling efficiency. Then put the case back together and get on with your life.
Possible step 3. If the power supply replacement doesn't fix it, you've got a motherboard problem. That could be as simple as a broken fan (some boards have a sensor to determine if the cooling fan is operating, and if not it shuts itself off to protect from overheating) or more complex, but easily repaired, loose connections, or a deeper failure of the motherboard. If you sense some activity upon clicking the switch but an immediate stop, then watch the fans to see if they spin up or not. Make sure all memory and peripheral cards are fully seated. If it still won't start, you'll have to consider failure of the motherboard.