For a PSU such as this where they used low quality electrolytic caps, just replace them all. Replacement electrolytics are inexpensive and it isn't worth the headache of trying to troubleshoot it again if another cap fails later. I have about a dozen or so PSUs in my work-queue that I'm going to be starting on in a few weeks and replacement electrolytics for them averaged between USD $12 and $25 per PSU.
As for replacement caps, you want low ESR high ripple current rated parts for the filter caps. I prefer Nichicon HE or HV for those but when a specific value/size isn't available in those two series, I'll go for Nichicon PW or PM. For the smaller parts (generally 47uF or smaller) I'll use Nichicon PW but very occasionally (0.33uF, 0.22uF, 0.1uF) I'll use VZ. The larger two main input filter capacitors don't fail as often, but I generally replace those as well. Nichicon CS, CY, GU, and similar series tend to be suitable replacements for those. Beware though, there are a /lot/ of counterfeit electrolytics on eBay, so stick to a major vendor such as Digi-Key/Mouser/Farnell/Newark/etc.
Not only do you need to match up the value (uF) and voltage of the capacitors though, but you need to make sure the replacement has at least the same or better ripple current rating and also match up the diameter, height, and lead spacing. Capacitor manufacturers specify the physical measurements in mm in their datasheets, but beware, sometimes a particular value/voltage capacitor is made in 2 or 3 different physical size combinations. Sometimes you can substitute a smaller/larger diameter (10mm for 12mm for example) or a shorter/taller part, but for larger diameter parts (8mm or larger) try to match the lead spacing so you won't stress the solder joints. In general though, with modern PSUs made within the last 10-15 years or so, you want to avoid replacing a larger diameter part with a smaller diameter part even if the lead spacing matches because the larger diameter part will have a higher ripple current rating and be rated for longer service life.
Another tip, make a list before you order anything. I've found it easiest to go with a format similar to:
C1 1500uF 16V 10mm 25mm 5mm
Where C1 is the board designation, then value, voltage, diameter, height, and lead spacing. Making a list such as this will save you a lot of headaches. As you add each part to your list, mark the top of each capacitor with felt-tip sharpie marker too.