Fastest way to lose height

This recent one is very much on topic:

Some skydive pilots also use a side slip to create extra drag allowing them to lower the nose more.
Gear is good to have out IF you can keep close to Vne with it. On some aircraft you are very limited when the gear is out. Same goes for flaps I guess.

Flaps extension is generally not recommended to my knowledge. If they are overspeeded, there is a of course a risk of structural damage. I have not seen hard data but I am convinced that maintaining Vfe with full flaps is rarely as effective in loosing height as diving Vne with flaps up. (Besides, the higher airspeed might blow the fire out.)
Side slips is on some types very useful to steepen a low-speed descent, but not to loose altitude generally, because use of full rudder/aileron is limited to maneuvering speed, Va. (However, in the case of an engine fire, I have seen side slips recommended to direct flames/smoke away from the cockpit).
Gear down is type dependent, and arguably also depending on whether there is an emergency. On the Bonanza, for example, the max gear extended speed can be overruled in an emergency and an extremely fast descent can be made at Vne with gear extended, risking damage to the gear doors but otherwise safe – this is per AFM.
Spiraling down helps a little because it increases induced drag, somewhat acting as a speed brake. But g-loads has to be limited to that permissible at high speed, so in an emergency with a lot on one’s mind I would probably avoid it.
On the Corvalis and other fixed gear types with speed brakes it is very simple: extend speed brakes, dive to Vne and manage the engine cooling according to whether there is an emergency or not.

Peter wrote:

The detail is likely to be type dependent but I wonder whether having the gear out (on a retractable) and/or flaps out is the best way. It may be that flying at the highest speed, close to Vne, in a spiral dive, may produce the fastest descent

In every pressurised aircraft I have flown there is a manufacturers procedure for emergency descent. Gear/no gear varies by type. I’ve never heard of the use of flaps or any sort of spiral descent, although there are rules for the RVSM airspace to consider. It’s frequently practiced in training

In the CJ4 there is an automatic emergency descent mode triggered if the cabin depressurises, which descends to 15,000 ft without exceeding Vne. All you have to do is pull the power levers back. Handy if you’re feeling short of breath at FL450

In the Diamonds, where Vle = Vne, the prescribed procedure is gear down then fly at Vne (Vno in turbulence). Personally, I would add steep turns as well.

The other way would be to throw the aircraft into a spin. :)

The generic emergency descent which would be expected in, say, an IR check ride in the USA (EASA seems to not bother with scenario based emergencies other than EFATO in a multi), is gear down, enter a 30 degree bank spiral dive at Vlo, and recover at a stated altitude. VSI is normally pegged in this configuration.

I once met a pilot who had just been flight testing an Isaacs Fury that was built with wing-wires a gauge too thick. Apparently it wouldn’t reach Vne in a vertical dive with the engine at full chat.

I was taught during FI training to favour the spiral descent with full flaps at Vfe after doing FIRE drills (_F_uel off, _I_gnition off, _R_adio mayday, _E_lectrics off).

The reason being that you arrived at your forced landing field low-key point without excessive kinetic energy, making the ensuing forced landing more manageable.

You get a lot of drag/lift if you dare to spin it !

In theory, it seems you will loose more energy by flying flaps and slow from turbulenece drag than fying clean and fast from speed drag with the risk of structural damage and max energy.

As Mark1 mentioned, the best I can think of is full flaps at VFE with 30 degress turn and side slip