Cameras have a variety of flash modes (e.g. Fill-in Flash, Slow Shutter Flash, High-Speed Sync, Rear Curtain Sync, Slow Sync Flash, etc..)
So, which are the flash modes that use the least power assuming that there is no EV compensation?
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Let's discuss what each of the terms mentioned in your question means.
Fill Flash: When there is enough overall light in the scene to take a picture, but there are shadows that need to be smoothed out, fill flash can be used to lighten the shadows. Even outdoors on a sunny day, if the sun is high overhead or behind your subject you can use fill flash to even out the light. The camera computes the shutter speed and/or aperture needed for the overall scene, then adds enough flash to fill in the shadows. You may need to adjust Flash Exposure Compensation to get the look you want. The power needed can be very little to full power, depending on the shooting conditions.
Slow Sync Flash (a/k/a Slow Shutter Flash): This is the counterpart of fill flash. When there is not enough light to take a good exposure but you want many areas of the scene to be lit. The camera computes the shutter speed needed to properly expose the background, the flash fires with just enough power to properly illuminate your subject. Shutter speed will generally be slow enough that the camera will need to be supported in some way to prevent blur caused by camera movement. The amount of power needed will vary, depending on the scene.
High Speed Sync: Each camera with a mechanical shutter has a speed that is the fastest it is capable to sync with a flash. It is usually around 1/200 to 1/250 sec, but can be much faster or slower depending on the camera. At speeds faster than this the second curtain of the shutter begins to close before the first curtain is completely open. The sensor (or film) is not being exposed all at the same time, but instead is being exposed from top to bottom (or side to side for most older film cameras) by the opening between the two curtains. The faster the shutter speed, the narrower the gap between the first and second curtain. Since an electric flash strobes at a very short duration, only the fraction of the sensor that is behind the slit between the two curtains will be exposed to the light from the flash, and the top and bottom of the frame will have dark bars across them. The solution when flash is needed at a high shutter speed is for the flash to fire a series of bursts while the curtains move across the sensor. This means the flash must fire several times in very quick succession. To have enough power for that many pulses of light, each one must be weaker than a single, high powered burst. Each pulse is dimmer, but because the flash is pulsing many times, the total power used is relatively high in most cases.
Second (Rear) Curtain Sync: Normally the flash fires as soon as the first curtain is completely open. The camera then waits until the set shutter speed has almost elapsed and then begins closing the second curtain. With Rear Curtain Sync, the flash waits and fires just before the second curtain begins to close. The effect this has on moving objects is to make the place they are at the end of the exposure their brightest spot in the frame. The classic example is of a car traveling forward at night. With normal first curtain sync, the flash would fire early in the exposure and the trails of the headlights would extend in front of the car. Using second curtain sync, the flash would fire almost at the end of the exposure and the car would be seen with the light trails behind it. Second Curtain Sync is most effective when used in conjunction with Slow Sync. The amount of power used varies according to the scene.
In each case described above the amount of power used by the flash is determined by how much light the camera needs to properly expose the scene. Any of them might need more or less power depending on the scene.
It's unclear to me if you are trying to conserve battery power, or are looking for a mode which will encourage your camera to use a low amount of light output even though it has no manual control.
If it is this latter that you want, it probably isn't the raw amount of power you want to control, but the amount of power relative to the other sources of light.
You might get this with a fill-flash mode which lets you dial in the relative levels, but on a simple camera there may not be an option. So, your best bet is to use slow sync flash in a mode which lets you control the shutter speed (like, most obviously, shutter priority mode). Set a longer exposure, and you'll have more ambient light and less relative flash.
It won't matter if you use rear curtain or front curtain (regular) slow sync; that simply determines when the flash fires relative to the shutter duration.
Assuming flash power is automatically controlled (no flash EV compensation), the flash will fire with less power the wider your aperture (although on a compact camera you're unlikely to have much control over this). That won't change the balance of light, but if you were concerned with using literally the least power, using a wider aperture (and higher ISO) will help.
So, which are the flash mode that use the least power assuming that there is no EV compensation?
It's not the flash mode that's going to make a difference -- most of the modes you listed deal more with when the flash activates, and the flash doesn't care whether it syncs with the front or rear curtain. What's important is how much light it produces with each activation.
Fill flash is meant to fill in the shadows rather than to illuminate the entire scene, and you often don't need full power to do that. High speed sync, as I understand it, also involves firing the flash at lower power, but it does so several times in quick succession. So, of the modes you listed, I'd suppose that fill mode is likely to consume the least power on average.
This question has been well enough answered by the other answers BUT the information has been divided between comments and answers. The object of this answer is to point out the single key difference between the two main situations that occur.
For a given level illumination of the subject by the flash with all other factors the same. ALL use about the same power except for High Speed Sync (HSS) flash.
This is because, as several people have said or implied, HSS flash fires multiple times, with each firing having to be at the same power level as a single flash for any of the other modes. Michael Clark explains it best (IMHO) but the answer may still confuse slightly.
As Michael says, in HSS mode the shutter is never fully open. Instead two "curtains" move across the image plane with a gap between them. If the gap is say 10% of the width of the image then the effective shutter speed is 1/10% = 10 x faster than the time that it takes the window to move across the image. This is because the light falls on any given area of sensor for 10% of the time. So if it takes say 0.005 of a second (= 5 ms = 1/200s) for the window to cross the sensor then the slot is fully over any given area for only 0.005/10 = 0.0005s (0.5 mS = 1/2000 s).
If the flash fired once in the above case it would illuminate only a single 10% wide slot of sensor. To cover the whole sensor area the flash will need to fire at least 10 times in the above case. BUT if it fures 10 times any misalignment in timing is liable to leave either bright edges or dark edges between 10% bands - so it will probably need to be fired somewhat more often than 10 times.
However, at each firing the whole image is illuminated to the required level but only 10% is viewed. So the flash uses 10 x as much power! Consequently, as shutter speed goes up the maximum flash intensity will drop due to the limited amount of energy available. A large flash reservoir capacitor will allow more flashes in the time available BUT this will place additional thermal stresses on the flash tube.
At the limit the HSS mode will use as much power as the flash can provide. At lower settings it will use more power than any single shot mode with the same flash illumination level, and as shutter speed goes up the required flash power will rise. Above a certain speed the flash will reach maximum power and the available flash level will then drop as shutter speed increases.
Some flashes (eg my Sony 5600 HS(D)) do not allow HSS mode if bounce mode is selected - as the power requirements are further increased by having to bounce.