I've found this schematic diagram for generating + and - 5V from a single DC supply.

From what I usually see, voltage regulators have bypass capacitors connected on the input and output sides.
However, I do not know where to connect the bypass capacitors in this dual supply case.
Can someone please tell me where to connect them and their values?

 

That circuit is wrong, it wont work.

You can get dual opposite voltages from AC but not DC.

 

I've found this schematic diagram for generating + and - 5V from a single DC supply.

This circuit outputs +15V and around 0V where there should be -15V. The input of the 7915 is the same as it's output reference.

Depending on the current you need there are various approches to create a ground. If you want +-15V from 24V it will need to be some kind of capacitor switcher/charge pump / step-up converter because 2x15V > 24V.

If the output voltage is lower than 24V then there are other possibilities. There are IC that create a negative voltage from a positive one like the LT1054, you can also create a virtual ground with optocouplers or transistors. What would be the best solution depends on how much current you need.

 

The label for the thread and the drawing have different voltages. Minor confusion.

You can do a +/- 5 DC out of 24 volts worth of batteries, but you need a "ground" in the middle of the batteries.

The capacitors go on the chips, from output to ground pin, and input to ground pin. Described in the datasheets for the chips.

Since 12 volt batteries are very common, I might assume you have 2 of them and all your ground leads in the drawing need to go between the batteries.

 

I discussed this in detail here...

Creating a Virtual Power Supply Ground