How can I safely test how a power supply handles abnormal situations?
What you want to perform are safety tests the quality department of the power supply manufacturer should have done before certifying the converter. Among the classical tests like short circuit or line dropout, there are the open/short tests which consist of opening or shorting a component and see how the power supply behaves in this case: if protection trips or the fuse burns peacefully, that is fine. If there is an explosion, a thermal runaway with risks of fire or any hazard for the user, the test fails. Adjacent pins or any pin to any pin short circuits on the control circuit are also performed and,  believe me, you often replace the controller and the semis on the board when you do it - after yelling of course : )

That is the reason why while reading electrical diagrams you often see redundant components like multiple series-parallel resistors or capacitors. This is to make sure opening or shortening one of them does not break the chain. You can imagine how painful it is to run these tests

Now, for your converter, a few advices:


a good practice is to perform tests through an 1:1 isolation transformer. If we talk about low-power converters without a power factor correction (PFC) front-end stage, you can even power the converter via a high-voltage dc source. That way, you can safely limit the output current and limit damages in case of problems.
wearing safety glasses is important but I rarely see them in labs. That being said, when a poor-quality electrochemical cap vents, don't put your eyes above and make sure ventilation is on. Now if you talk about big converters, these glasses are mandatory.
To check robustness against overload, slowly increase the output current and see how the converter reacts when you exceed the maximum value. Do it at both lines extremes: for a 100-240-V universal input specified on the enclosure, the normalized min and max are 85 V rms and 265 V rms, sometimes higher.
A classical issue is over power protection or OPP: for a 2-A power supply, the limit will be 2.5 A at low line for instance but can easily be 3 or even 3.5 A for a badly-compensated converter powered at the maximum input. This is something you check in the above test.
Over voltage protection or OVP: in case the loop fails, the power supply either latches off completely or goes through an auto-recovery hiccup mode (you can hear the "tic" "tic" noise). To test if this works, simply short the optocoupler LED in the secondary side while the power supply runs and observe the output voltage. The overshoot should be less than the voltage rating of the capacitors of course but \$V_{out}\$ should also be constrained below a safe value, e.g. 5.5-6 V for a 5-V nominal or 13-14 V for a 12-V output as an example. If the output goes to 10 V for a 5-V output, not good. If you wanna play, you can short the LED at the highest input voltage: if the power supply is well designed, it should be ok.
short circuit protection: you can apply a short circuit (thick wire strap) across the connecting terminals and not after the long cables via the electronic load which would still ensure a bit of voltage at the board terminals. The short circuit at the board connectors level is a worst-case as the converter should enter a deep continuous conduction mode (CCM) without voltage reflection (\$V_{out}\$ is truly 0 V). Stress is maximum in the secondary and primary sides. Again, short circuit at the highest input is the hardest test to run. You can also install a short circuit and start the power supply multiple times.
Finally, if you wanna go deeper in the tests, you can probe voltages and currents and see how they compare in nominal and extreme conditions with the maximum ratings of passive and active components. For instance, serious designers derate the semis: 10 to 20% voltage derating on a high-voltage MOSFET. If the \$BV_{dss}\$ is 600 V, you'll make sure that in the worst possible situation (hard short, OVP, surge etc.) the voltage across the transistor will remain below 0.85 x 600 = 510 V. For a diode, you will first adopt a 100% derating (200 V diode for a 100-V max spikes) but once snubbers or dampers are installed and checked in all conditions, a 50% derating may often be ok. 


A typical failure for caps is overtemperature by having too high a rms current circulating (I see that a lot) and designers are tempted (or imposed by the buying department) to use cheap capacitors and they won't last long even if the rest of the converter is sound. Check the brand and it they ain't coming from serious (known) manufacturers, it is likely to be a cheap ready-to-fail type.

This is a short list of what can be done for testing a converter and there are plenty of other procedures like on-off cycling, burn-in at full load etc. that serious power supply manufacturers follow. But if you already check that over current protection, OVP and other protections I described in the bullets are ok, it is a first good indication that the power supply is properly designed.