The definition of a fuse is to protect equipment against overcurrent and the following damages, which usually can result in fire and nasty smoke from burning insulators like rubber or PVC.
There are several types of fuses tailored towards the equipment, that has to be protected.
The most common types of fuses we see in our daily life are fuses (and breakers) that are intended to protect the cabling/wiring.
In the distribution box coming from the main distribution/street transformer are fuses or breakers that are suited for a certain kind of two kinds of overcurrent situation:
- slight overload, like too many devices pulling energy at the same time on the same outlet
- short circuit somewhere, so a sharp spike of current has to be dealt with
The energy needed to trip the breaker/fuse is coordinated with the cable (diameter), so the electrician does usually not need to calculate complicated stuff.
In germany, where VDE standards take place, usually it is something like: 1.5 mm² will be able to carry 16A, so use a fuse 16A gG, or a breaker 16A characteristic B- if there are lots of cables in the same duct, or in an isolated wand, some reduction factors apply.
Usually fuses should be "coordinated"- this means, that when they should act, only the fuse that is next to the problem should act.
Fuses (and breakers) also have a maximum breaking capacity, so when a short-circuit situation arises and a small fuse is not able to really switch off in time, the next upstream fuse will also blow.
This said, a fuse on the primary of a PSU usually will protect against slight overload. In a short-circuit situation I would suspect that the fuse/breaker of the wall socket/circuit also has a good chance to act, depending on the short-circuit-current produced. (So called "backup protection")
For protection of sensitive (and expensive) semiconductors, there are specialized fuses- fast acting and able to take high short circuit currents- in germany there is the aR/gR line of fuses for mains distribution, and for parts within appliances usually fuses class "FF" would come to mind, that are able to break DC currents without problems.
In this case, you would have to look for your sensitive portions of the device, have a look at the short circuit currents which could occur at the problematic points and have a look how fast those current would overheat your part.
From this value (current and time) you calculate the integral value I²t in A²s, and look into the datasheet of the fuse in question- and the voltage drop of the fuse...
Then you look if this would help your circuit or if protection can be achieved in other regards.