SMPSU's aren't good things to repair, for a start they are full of high voltages and high frequencies, so quite hazardous, the primary side is also live to the mains, so an extra shock hazard, and it causes problems for using an oscilloscope.
A few common issues:
1) Switch-modes supplies are sort of 'magic', they generate their own power to run themselves. But they can't do this without power in the first place, so they require some kind of startup circuit - years back this was often an electrolytic capacitor which gave a pulse of energy to 'kick it' in to life (these go faulty, and the PSU won't start up). But in more modern times a high value startup resistor is commonly used, again these fail (they go high or O/C) and the PSU won't start.
2) You can get short circuits on the outputs, often the rectifiers go short - this causes the PSU to shutdown to protect itself.
3) Electrolytics going high ESR, this is by far the most common fault in all electronic failures, 'sometimes' you can see they are swollen or leaking, but often there's no physical sign - the ones in the picture 'look' OK.
For a start see if you can read the number off U10, and if you can then try and download a datasheet for it - that will give you an example circuit, which should be near enough to what you have.