Due to the many ill effects of hard water on our everyday tap water use at home, it is sometimes necessary to put a water softening process to take care of this problem. In the case of home aquariums, however, softened water may induce bad health cases for your fishes, but of course, results still vary from a case to case basis.
For one, softening water works by changing the chemical composition of your hard water by reducing magnesium and calcium deposits in the water and exchanging them with mostly-beneficial sodium ions. Now, I said this is mostly beneficial only because it helps with non-aquarium uses, but in the case of home aquariums, generally, you would not want any change in the chemistry of the water where your fishes reside.
Most fishes are very reactive to changes in water acidity and are not suited to live in conditions where the water is tested with a very low pH level. Softening water is basically lowering the pH level of your natural tap water after several chemical reactions to compensate with its hardness. With this fact alone, it is already not advisable to expose the fishes in your home aquariums to softened water if you are worried about their safety, and just like any other fish lover, who would not be worried, right?
Anyway, even with this knowledge, many people will still counter that they have been using softened water for their fish tanks without noticing any adverse effects on their livestock. This may have been caused by a lot of factors. Maybe their fishes are already accustomed to that particular water environment from the time they are bred in captivity, or maybe because the pH drop is not that significant to provide any great effect to the health of the fishes, but in any case, what works for them may not work for your aquariums too. So if you are in doubt about this subject, it is always advisable to stay on the safe side and not use softened water for your own home aquariums and keep on using the same water you have been using for your fishes since the day you kept them.