Fuses are incorporated in the design of equipment to protect critical parts against excessive current. A blown fuse can result from abnormal electrical surges or from issues within the equipment itself. I would contact the manufacturer if I had equipment that was frequently blowing fuses.
Regarding the type of fuse, most of my amps have come with fast blow fuses. One amp designer told me, in their opinion, slow blow fuses were not "as linear" as fast blow fuses so they did not believe the slow blow fuses to be as good sonically.
Regarding quality, the past few years have seen several manufacturers using different (improved?) materials in their fuses, from gold end caps, to ceramic bodies, and even different metals such as silver or copper for the filament. I suspect the connection of the fuse to the circuit board could also have some effect on sonics. Some folks wrap teflon tape around the fuse or around the fuse and connection tabs to dampen resonances. Manufacturers of "high end" fuses include HiFi Tuning, Isoclean, Furutech, PS Audio and others. As an example, the PS Audio "Critical Link" fuse offers the following;
Direct gold plated end caps
OFC copper end caps
OFC copper fuse element
Foam damping inside
Cryogenically treated
One amplifier designer/manufacturer I know has high regard for the Furutech fuses on a sonic basis, and uses them in amps and preamps, even though they only come in a slow blow type. Another rather large manufacturer of high end gear told me within the past year they had not yet concluded how to approach aftermarket fuses with regard to equipment repair and warranty, and they had some concern that not all the aftermarket fuses were properly constructed from the standpoint of protection of the equipment. I found that an interesting comment.
