Selecting the correct input/output capacitors for a 7805

Obey the datasheet, but there is no harm in adding extra capacitors.

The 330nF and 100nF (non-electrolytic) capacitors are probably required to guarantee that the regulator is stable. They should be as close to the regulator as possible.

The 10µF electrolytics suggested on the website may be beneficial to the rest of the circuit. Eg. ripple smoothing on input and a "circuit-wide" decoupling on output.

So, I'd suggest combining the two. (Don't use just the electrolytics as they have strong parasitic properties.) If space is limited, I'd go with what the datasheet says only.

^{simulate this circuit – Schematic created using CircuitLab}

The datasheet does not recommend any specific capacitors, it simply mentions that the measurements were taken with 330nF and 100nF.

IMO this is a shortcoming of the datasheet.

If the regulator is stable without capacitor, then it should be written. If a specific value/ESR is mandatory, then it should also be written.

A cap at the input usually helps stability, as regulators tend to dislike inductive supplies. If the main supply caps are more than a few cm away, adding the 330nF cap mentioned in the datasheet would be a good idea, or any small value modern aluminium electrolytic.

Now, the output.

Considering the history of the 7805 regulator, I would pair it with a decoupling scheme matching its age, like a 10-100µF aluminium capacitor with ESR between 0.5 and 1 ohms, and a 100nF decoupling cap close to the load. I would avoid low-ESR caps.

If you guys are interested, I might be motivated enough to test one with the network analyzer.