Power supply design - what safety mechanisms are needed?
It might differ outside of Europe, but the following could be used as some basic overview of how to evaluate and select the standards needed, before comercializing a power supply.
Before starting with the design process you should define the following:

Customer / Field (e.g. medical, industrial, etc)
Location to be used (e.g. country)
Standards

Depending where and who you final customer is, the requirements and respective tests might change.
As an example, let's say you are designing a medical-graded open-frame 40W power supply, to be comercialized in Europe.
Until recently the IEC/EN 60950 (devices in the field of information and communication technology and for devices in the audio/video sector) was the basic standard that most of the electronics devices had to comply with before going to market. However, it was replaced in 2020 with the new IEC/EN 62368 standard. In some cases, for example industrial power supplies, you could even use the IEC/EN 61010 instead.
I am not familiar with medical-graded devices, but since it is the case it should also comply with the standard IEC/EN 60601 (Medical Electrical Equipment).
In order to ensure that your device is neither affecting or being affected by electromagnetic disturbances, it should also fulfill the requirements given by the standard IEC/EN 61000.
In summary:

EN 62368 (Basic Safety Requirements)
EN 60601 (Medical Safety Requirements)
EN 61000 (Electromagnetic Compatibility)

As for the tests, you should read the applicable sections of each standard and define a test plan accordingly. Next step would be to discuss with a standard expert, and a certification body since they will be the one conducting the final test on your power supply.
As you can see the standard and test selection can get quite complex, and that's why there are experts in this field, who also make sure that all the currently certified devices are still valid and up-to-date with the standards. Here are some additional design requirements / tests for your list:

Minimum creepage / clerance
Minimum hold-up cycles (AC)
Maximum allowed energy stored in the capacitors
Ratio of volume to capacitance
Exposed energized pins
Mechanical / Electronic Interlocking
Maximum stored energy
Overload / short-circuit protection etc ...

Although not 100% guaranteed, you can have a look at the certifications of some similar off-the-shelf power supplies. For example, this medical-graded power supply complies exactly with the standards mentioned above in adition to EN 55011. This should give you a good indication whether you are well covered.
This is a basic example and should not be taken as any sort of final solution. The best thing to do, is to consult a standard expert and a certification body, who will inform your exactly what you need.