Easily done.
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Fairly recent Nikon DSLRs (D700, D800) have an effective ‘sensor shaker’ which can be set to operate when the camera is powered on or off and helps remove all but the most stubborn dust particles from the sensor. This is particularly important with Nikon lenses which are poorly dust sealed – such as the the 16-35mm and 28-300mm AF-S versions which I use. These pump mighty quantities of air into the camera’s innards, along with any airborne dust, when the zoom ring is operated. It’s so bad you can feel the air rushing in if you remove the lens and zoom it close to your cheek. Quite why Nikon does not have its lens designers vent this blast of air to the outside beats me.
The Leica M10 has no sensor shaker, possibly because the compact body is already so packed with electronics and mechanical parts that there is no room for one. It’s probably the same reason that precludes installation of an IBIS system, which would be nice to have. But Leica has a clever workaround when it comes to sensor dust detection and removal. Go to the last page of the Main Menu on the LCD and click on ‘Sensor Cleaning->Dust Detection’. You will be directed to mount a lens stopped down to f/16 or f/22, defocused and pointed at a plain evenly lit surface. I used the 21mm Color-Skopar at f/22, focused on infinity, with my test wall just inches from the camera. Take a picture and you get this on the LCD screen:

Sensor dust disclosed. Click the image for a (yecch!) larger one.
Quite a bit of dust, something which can become visible in large plain areas in images, like expanses of sky. The picture on the LCD screen is rendered in the same orientation as the camera, as the red lens mounting index at left indicates.
Now go to Main Menu->Sensor-Cleaning->Open Shutter, first making sure your battery is fully charged. You do not want the shutter to close for lack of power when you are poking around in there. If the battery charge is below 40% a warning message requesting the battery be recharged will appear. Holding the camera upside down, LCD to the ceiling, blow in some air using a rubber blower bulb, directed at the sensor, being sure not to touch the surface protective glass, and redo the sensor dust image. Do not use compressed gas of any sort. The goal is to loosen dust particles so they can drop off, not blast them further into the innards of the camera. I got this:

Sensor dust gone.
Nice implementation by Leica and very easy to work with. The sensor is now clean as a whistle. If things had not improved I would have cleaned the sensor with an antistatic brush. Had that failed I would have resorted to a wet cleaning solution. I have used this product with success, and it leaves no residue, but despite the listing it does not come with a microfiber cloth, so make sure you have one. I cut a business card in half lengthwise and wrap the cloth around it. This makes for a flexible ‘wand’ and I spritz the tip of the cloth a couple of times, no more – you want moist not wet – with the solution and gently swipe the sensor’s cover glass this way and that. (I avoid Q-tips, finding them far too inflexible, meaning they risk damage to the protective glass on the sensor). Then a couple of puffs of air from the rubber bulb and you are done. This works for any digital sensor, not just the one in Leicas!
Because Leica M mount lenses do not have a zoom feature (the relatively benign variable focal length feature of the two Tri-Elmar lenses notwithstanding) the need for such sensor cleaning should be fairly rare. Further, when the lens is removed for changing, the sensor is protected by the shutter blades. So it’s not that easy for dust to get in there.
I had not checked for sensor dust since buying the camera second-hand 3 weeks ago, so only just got around to this, after noticing an out of focus blob or two in large smooth tone areas in LRc in this image, where the small aperture and short focal length of the lens emphasized dirt on the sensor.
If the surface you use to take the dust image is not evenly lit you will get a message ‘inhomogeneous (sic) image’. I got this more times than seemed right, despite using a flat, evenly lit, wall. Removing and reinserting the battery cured the issue.