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As the title says, plus how do you carry it and avoid the inevitability of landing on it and destroying it/yourself?
I used to. A dslr, then a m4/3. Crashed with/on both on a belt. Both I and the camera survived several crashes. So I guess you don't necessarily need to avoid it.
I do, on occasion bring my DSLR, I carry it in a mind shift rotation trail backpack, works really well.
I am yet to crash with it on, and no doubt there is 'more' risk that you will land badly on your back where the camera is and it would make any such crash work, I suppose its about what level of risk is acceptable to you.
I like to think I take it a bit easier when I have my camera with me... maybe I do!
I used to carry a Nikon D7000 in the top of my Camelbak, never damaged it. It was right at the top so I could whip it out asap to catch the riders behind me, so never in a case or under any padding.
I occasionally carry a Canon EOS 100D with a pancake lens in my pack. Occasionally a longer lens as well. It's actually not much bigger than a compact, which is one of the reason I bought it. Sits in a Lowe Pro case. I'm more worried about it getting wet than bashed to be honest. If it's showery I stick the whole thing in a dry-bag.
I find I used a waterproof tough compact more though. That lived in a small Berghaus accessory case on the side of the pack belt. These days I don't really bother with either, just pull out the phone for the occasional snap.
I used to occasionally carry a D80 in a chest pack clipped on to camelback shoulder straps. Now I just use a compact canon (sx720)
I do on big/scenic days out, i have a lower pro bag which is padded but also has space for hydration etc.
My Canon G7x mk11 (not a DSLR though) fits snugly into one of these cases :

not great for quick draw maybe, but solidly protected.
https://www.amazon.co.uk/gp/product/B001CCZF6E/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&psc=1
I keep my Sony RX100 (compact) in a Wildcat Cheetah bag at the top tube/seat tube junction. The bag has a small amount of padding so can take some knocks but it's also pretty much out of the way so unlikely to get a big direct hit. Easy to get out, take a shot and put back even when riding (easy terrain).
I do. Panasonic G9 with a bunch of lenses and flash. I use a Lowepro whistler 350 bag, it’s great and there is space for gear too. I have a Capture Pro clip on the strap for quick access and it works great. I don’t do that all the time but ride a lot like that.
Sometimes I will take the kit out with me but make sure to ride within limits if I do - Expensive mistake if not! DSLR body and a single lens should pack away in the middle of a decent sized camelbak and then just wrap it in a towel or similar
I basically don't take my DSLR anywhere these days since my iPhone (X) can take better photos 95% of the time. The other 5%, well, I can live without for the convenience. I can't remember the last time I used my DSLR. YMMV.
I sometimes carry my Olympus EM5 with one of its smaller lenses (17mm prime, 12-42 pancake zoom) in my Camelback. Landing hard on my back is sufficiently rare that the camera has survived so far.
Background to this is that I was thinking of doing a few selfies using cheap yongnuo flash/shutter triggers and somehow mounting one to the bars. Similar to something I saw Doddy from GMBN doing on an old post of his only he was using the pricey pocketwizards. You can get great results!
I occasionally carry a M43 and small prime lens. That's less and less though as phone cameras have got so good.
Gopro handlebar clamp with a hotshoe mount then attach the flashgun to that?
I’m in the same frame of mind as superficial above.
Modern smartphones are so capable they render a DSLR redundant for most users, most of the time.
However in answer to OP, i use a lowe alpine slingshot bag, really convenient. But only occasionally, rest of the time i use an iphone x.
‘Best camera in the world is the one you have with you’
Full size DSLR and a few lenses for me on occasion in an EVOC Photo Scout bag, often with a tripod. Not worth it unless I was cycling to do photography. Cycling and carrying stuff to record a ride I’ll use anything from iPad to phone to old compact.
There are loads of cheap used compacts that are good enough to take on the bike and cheap enough not to worry about. Canon G7 onwards are great. I bought a Sony Cybershot H10 for £15 its very small with a big zoom. (iPhone x will probably take better pics though in most cases though)
‘Best camera in the world is the one you have with you’
But if you have a choice of cameras why not take the “best” with you.
I have used an Alpkit Stem Cell to carry a small Micro 4/3 mirrorless camera (Lumix GX800). It is handy as easy to reach and get camera out and take a photo, while on the bike. Not tried it for much proper mountain biking - would be worried about vibration etc, maybe better in a backpack.
+1 for the Mindshift Trail.
Use it to carry my XT1 with 18-135. Very quick to whip it out if you take the lens hood & strap off.
I tried a Peak Designs capture clip on the shoulder of my pack. Convenient, but vulnerable while the Mindshift has handled the odd OTB. It ain't light though.
Current thing is a s/h olympus xz2 compact off of ebay (fast zoom, nd filter built in, slightly larger sensor, good controls, shoots raw). I'd have a sony rx100 ideally that does what I want but a bit expensive for something that I'll inevitably break (have a box of bust trail compacts that lasted a few years before knocks got to them).
I did try for a bit 'phone only' but I need zoom for framing, and better quality in the often poor light that you get in the hills of Scotland, once you start zooming or processing raw in lightroom, mobiles aren't up to it
I normally store the camera in a front pocket of either a mountain hardwear singletrack vest / montane razor pack, close to hand, not getting the same vibration as it would on frame, seems to be in an area relatively immune from crash impact
I do as CTK - cheap compact just kept in a pocket.
Used to carry a cheap rugged camera in a Lowe bag strapped to the frame, but now just my old phone which has a stupidly good camera for its age (Oneplus Two). Only thing I occasionally miss is a reasonable optical zoom, but for me that's outweighed currently by having a lofi/retro camera app on the phone to minimise post processing.
Panasonic TZ70, great compact camera that fits in jersey pocket
When I've carried my SLR it's either been stuck in my Camelbak wrapped up in a jersey, or I also used a ThinkTank Digital Holster chest mount for a while; more convenient than the Camelbak.
As for phones being better than DSLR....that must be why all the pro photographers use phones these days....!
SLR has been out many times on the bike and also when skiing. I just shove it in a Camelbak HAWG NV, with other stuff, spare lens even sometimes and a bit of stuff to pad between things but don't bother with loads of padding as the bag is fairly padded anyway.
Fine, and have taken a number of tumbles with it too. The camera is old (Nikon D70) and taken beating over the years but only a bit chipped off the flash cover in all that time.
However it's come out a lot less because yes, the phone takes cracking shots too.
Sure, it can't match the lenses for zoom, clarity and depth of field, so I miss the SLR for that, but the vast majority of shots I'll take riding are just a bit of scenery these days and arty static shot of the bike somewhere scenic.
I used to try to take action shots and all that but then it's way too much faff setting it all up and myself and friends are too impatient for that. Can't be stopping frequently on rides and faffing about setting up shots, sessioning to get the best one etc. Just want to ride.
Did have a compact zoom which was okay but it got bashed up far worse than the SLR.
As for phones being better than DSLR….that must be why all the pro photographers use phones these days….!
This still makes me chuckle:
https://www.theverge.com/2019/3/11/18260213/huawei-fake-photos-p30-pro-camera-ad-dslr
Nikon D5600 with 18 55 kit lens and a Capture Clip like Doug uses (waves!👍).
Does the job perfectly. Mag quality images for under £500! The next cover is one I took with it so no need to spend a fortune on expensive kit!😀
I used to take my Nikon D40 & kit lens with me,wrapped it in a jiffy bag & kept in the Camelbak.
These days though I usually take a Panasonic Lumix or use my smartphone.