"It depends" is the best answer. We need to know more about the applications, especially voltage and current.
Some thoughts that might be helpful.
You need to know if you need AC or DC. Obviously you are replacing batteries but if the batteries are powering filaments then either AC or DC might work.
You need to know the voltage. For many applications the exact voltage isn't that critical - plus or minus 10%. What I'd avoid doing - using a 12 volt supply where you need 9 volts or 6 volts - or 15 volts. It might or might not work but worse, it might damage what you are powering.
You need to have some idea of the current requirements. There is no great precision required here except that the power supply must be capable of supplying more current than the load requires.
Regulated or unregulated does mean something. A regulated supply is likely to cost more money but will work even where unregulated power is required. It might be overkill but the lack of unregulation is not likely to be a problem. The reverse is not true. If your load requires regulation and you provide unregulated power it might or might not work.
Linear, switching or other modes refer to the method by which the voltage is regulated or controlled. Linear power supplies are often thought to be simpler and quieter (RF) than switching but there are some switching supplies out there that are quite simple and quiet. Linear might not be as efficient, especially at higher powers, so more heat might result.
If your needs are DC and you know the highest voltage/current you'll require you can purchase a well regulated supply to meet those needs then as suggested, add step-down regulators. You will need a large enough difference between the power supply voltage and the voltages below to allow regulators to work. In this arrangement you'll be tied to the base power supply - often quite acceptable on the bench but cumbersome otherwise. Note that interaction between loads is possible if they are noisy or sensitive.
Note that you need to address overcurrent protection in the form of electronic limiting as well as fuse or circuit breaker. You need to protect the power supply but also make sure that you don't damage what you are powering or the wiring to it - worse yet cause a fire or injury. The mains side requires a good deal of caution as well.
Some things may be affected by the elimination of batteries and connection to external supplies. The power leads can act as antennas or destabilize a sensitive circuit. There are remedies for this but it's something you should know.