Everyone has different needs and values different things. If you're looking for the "best" at something, you often have to sort through mountains of other information, so I'm going to take a stab at maintaining a page with some of the more important factors people look at.
Before proceeding, let me first put on my bullet-proof vest and then my armor system on top of that, then crouch behind the three foot walls of my bunker…
- Best Autofocus. The Nikon 1 is still hard to beat in decent light (it struggles a bit in very low light). The phase detect system embedded in its image sensor is wicked fast, and it has better follow focus characteristics than the contrast-based systems most of the other mirrorless cameras use, and even the few that have phase detect on sensor. Caveat: it takes some study to master the system.
- Best Low Light. The Sony A7sII is the clear winner here, with the A7II and A7rII also quite good. The advantage of full frame and those Sony Exmor sensors can’t be denied. Caveat: lens choice is still somewhat minimal as I write this, especially if you want fast zooms.
- Best Dynamic Range. A log-jam if approached purely from measurements. For crop sensors at base ISO you've got three strong contenders for wide dynamic range: Fujifilm X-Pro2, X-T1, Olympus E-M5II/E-M1/Pen-F, and Sony A5100 and A6300. But overall, the Sony A7 models changed the game here and the Mark II models pushed that up some more: they’ve got better dynamic range and now we can get uncompressed raw files, too. I personally would contend that at base ISO you'd be quibbling over unimportant nits between most cameras, and that we've got plenty of dynamic range with most recent mirrorless cameras. While "dynamic range" is often cited by people as the reason they favor camera X over camera Y, in reality I think this is chasing the wrong suspect. Even more curious is how many people who say they want more dynamic range also select higher contrast settings or post process to add contrast, which tends to remove all that extra captured information. Make sure you really know why you want and need more dynamic range before using this as a criteria for evaluating cameras. Caveat: dynamic range is measured and evaluated differently by virtually everyone, and modern cameras now do slightly different things at higher ISO values that may change your decision.
- Best Resolution. The Sony A7r changed the game, and the A7rII pushed that game further to 42mp. It’s currently the highest resolution mirrorless camera by a fair margin. The Olympus cameras have a mode that allows them to take high resolution images on a tripod by moving the sensor for eight continuous images that are then converted into a single image file. This, too, produces very high resolution, but I can’t consider it “best” because it’s not useful if there’s any camera or subject motion. Caveat: watch out for diffraction, which will tend to visually steal back some of the resolution you may capture.
- Best Video. For awhile this was a Sony no-brainer for consumers, GH4 for the pros. But because the HD TV bar isn't set very high and everyone has been adding more video capability with each generation, it's not so clear cut any more, at least for consumers. I suspect that the "convenience" of video is now more important than particular performance aspects for most people. Thus, the Fujifilm X-Pro2 and X-T1 are not winners because video is buried on those cameras and has some impacts from the X-Trans sensor use, while many of the other contenders make recording video easier. For outright capability, the best choice is still probably the Panasonic GH model, currently the GH4. Note that we now have a video-only player, the Blackmagic Design Pocket Cinema Camera, which also warrants a close look. If you don't need stills, it might be the best choice yet. If you’re talking 4K video, the GH4 and Sony A7sII are probably the best choices at the moment. Caveat: a lot of the smaller cameras overheat when doing video for any length of time, most have time restrictions on clips you want to be aware of, and all of them have fairly poor microphone amplifiers in them.
- Best JPEGs. Probably the most subjective category of "bests," as we all have different preferences in terms of color, contrast, and more. Heck, many (if not most) of the males reading this have some form of color blindness, which will certainly impact how you evaluate an out-of-camera JPEG. That said, the Olympus mirrorless cameras have consistently excellent JPEGs in my experience, especially if you take the time to tweak the defaults a bit. The Fujifilm X-T1 (and other X models) also produce superb JPEGs. Caveat: but those nice looking JPEGs from the Olympus and Fujifilm are not color, saturation, or contrast neutral. Moreover, virtually all the cameras allow you to tweak the look of JPEGs in ways that would allow you to get a look you like. Technically speaking, I don't think there's a best here unless you're lazy.
- Best Lenses. The m4/3 system wins hands down, due to both choice and quality. Fujifilm gets a strong nod of appreciation for producing excellent lenses with a reasonable choice. But Fujifilm still needs more choice, especially at the telephoto end. In my estimation, Sony is mostly let down by their lenses, particularly the zooms (with a few exceptions). Samsung also gets a slight nod for producing some very nice options (the 16mm f/2.5 and 85mm f/1.4 come to mind, as do the fast zooms), but like Sony there are many weaknesses across the entire range. If you're one of the 1%, Leica would clearly take the cake for lenses here, but I'm pretty sure that less than 1% of you reading this are in the 1% ;~). Caveat: this is a "best" that is in constant flux, and if you have a particular focal length preference (or preferences), you may be best served by one mount over another. For example, the Zeiss 24mm for the NEX is about the best 35mm equivalent option out there, and clearly superior to the Olympus 17mm, but the 35mm FE-mount Sony Zeiss outclasses both.
To fully appreciate the "bests" we also have to consider the "worsts." As you're about to see, we've got products that are best at something and worst at something else:
- Worst Autofocus. I nominate the Canon EOS M3 and M10, which despite being fine cameras are a large disappointment in the focus department. There are compact cameras that perform better.
- Worst Low Light. I'd tend to award this prize to most of the earlier 12mp Panasonic m4/3 models (the GH2 is somewhat better than the rest; the Olympus Pen-F is currently the state of the art for m4/3). But you also have a lot of fast lens choices in m4/3, so you can mitigate that problem. While some people complain about low light performance, we're still far better off than we were with film. Unless you're shooting in low light for a living, I wouldn't overly obsess with the "worst" in this category. If you're surprised that the Nikon 1 isn't the winner here, it's because Nikon managed to pull a rabbit out of its hat: the Nikon 1 swings above its sensor size expectation, especially the new J5. While that's not quite up to the best m4/3 sensors, it's close enough not to ding it here. The bottom of the heap is currently the Ricoh Q series cameras, by the way, mainly because they use such a small sensor (smaller than 1”).
- Worst Dynamic Range. Probably the Nikon 1 and Pentax Q. But in pure measurement terms we're talking about a difference of maybe 18% with the Nikon J5 compared to some larger sensors (not full frame). Considering its small sensor, the Nikon J5 is actually performing a bit above expectations, as I just noted in the low light section. I should point out that, of the current cameras, the Pentax Q series is clearly the measured worst performer here. But it's trading size against image quality, by using a compact camera sensor.
- Worst Resolution. Depends upon whether we're measuring absolute resolution of a capture or theoretical resolution, plus what our output choice is. Technically most people are talking about sharpness or acuity, not actual resolution, when they use the term "resolution." I'd tend to give all the cameras a pass on this category for most normal uses. It's really only if you're going to print big that resolution is a big issue, in which case avoid any of the cameras that can’t give you a minimum of 16mp.
- Worst Video. The Leica M9, which has none ;~). Leica did update the M to include video, but you can still buy a M9 variation without it. Still, a number of cameras don't have a great deal of flexibility in their video features. Thus, you have to make sure that the camera you choose can actually perform the video format choice you desire.
- Worst JPEGs. I’d tend to say the Leica M's, surprisingly. Everyone else has flexible enough controls that I'm pretty sure you can find some setting that'll produce what you want.
- Worst Lenses. As time passed, I had to change my answer here: pretty much everyone is making good to excellent lenses, though Sony still probably has more weaker zooms than the rest of the players in their APS offerings. The real issue here is quantity of lenses. m4/3 wins hands down, while everyone else is still playing catch up.