I am working on a project that will be permanently installed and powered in my car. I know the Arduino has the on board power regulator, but the voltages put out by the cars alternator can reach 13.8V or so.
If only that were true - the automotive environment can be harsh, electronically speaking, as all sorts of
spikes and drop-outs on the nominal 12V can happen - a car contains dozens of electric motors relays and solenoids, any one of which may be misbehaving and injecting inductive spikes into the wiring
harness, and they all inject considerable noise when working well.
Assume 30V+ spikes are possible, and that the supply can drop to 6V or less when starting the engine.
I reckon that means using some noise-filtering, and an external voltage regulator with a high input voltage tolerance is wise. LC filtering can be pretty tricky to design right, RC filtering is easier.
Say your Arduino circuit uses a maximum of 100mA in total. You could feed the 12V supply via a 10 ohm
resistor and 100uF 25V rated electrolytic to a 7810 or 7808 regulator, and from that to the Arduino jack.
The RC combination should tame spikes and short drop-outs, the external regulator will share heat dissipation
so the Arduino regulator isn't heavily loaded. The 10 ohm resistor will dissipate at most 0.1W which is safe.
Adding a fuse on the input 12V is very wise, either a 20mm glass fuse of suitably low rating, or a polyfuse.
The fuse is to prevent wiring fire if there's a short circuit, it might or might not help to protect the
Arduino but usually semiconductors melt before fuse wires! However the 10 ohm resistor itself is likely
to act as a fuse as it will dissipate 15W on a short circuit.
As mentioned above switch mode supply is a good approach too, it will naturally protect to some
extent against drop outs and spikes too. TVS diodes can be used also to protect against transients.