I'll just dump words, and put all data at the end to support my claims as much as I can.
Mainland
The most common input editor by far on the mainland is pinyin input. Sougou, Windows or Mac's native IME, google's IME (which had an incident of plagiarizing sougou's database), QQ Pinyin, Baidu pinyin etc. For people not satisfied by regular Quanyin (whole) pinyin, there's Shuangpin (double pin), in which you only type 2 keys for any character, the initial consonant and final section (e.g. Lai = L + L, as L maps "L" in the initial position, and "ai" in the final position). This is the one I use, as I like speed and efficiency in typing, and I'm willing to remember the positions of the finals on the keyboard (which most people find slightly harder). With a good IME, like sougou, there's no doubt that this is the fastest input method, with only 2 strokes for any character, or 4 for 2-character compounds.
This is by far the most common. After this, some specialist typists like to input by stroke, for which the most common on the mainland is Wubixing. This inputs by shape of characters. For instance, 胡 is D(古) + E(月). The steep learning curve deters most people, though, and truthfully it's not faster than Shuangpin. However, it's possibly the only input method on the mainland that maps every character uniquely, so as long as you don't make typos, you'll never need to choose characters from a dropdown list. Also, most 2字 compounds are mapped to 4 keystrokes (first 2 of each 字).
Finally, on phones, quite a few people like to use Wubihua (not our previous wubixing). Users type the strokes of the character in question from 横，竖，撇，捺，and 折. Some people take this out and just draw characters on their touchscreen, and others just stick to pinyin all the time.
Truthfully, I dislike pinyin input on phones, because either the keys are too small on the keyboard (and I miss, then have to backspace and retype, or deal with a list of fuzzy characters to choose from), or are 3x4, which just makes the list of characters to choose from way too big for me. On smartphones, somehow just writing the characters (or using stroke input) seems to be the fastest (unfortunately, considering the effort needed to use either of these)
Taiwan
The most common IME for computers here is Zhuyin fuhao（注音符號）, also known as bopomofo. This uses the taiwanese phonetic alphabet for chinese to input characters, tones included. It doesn't map 1:1 with pinyin, and tones remove ambiguity (and need to type space), so it's probably slightly faster than Quanpin. On phones, people sometimes use a traditional variant of wubixing.
As for shape input, Cangjie （倉頡）is common, and to a lesser extent Dayi (大易). Cangjie is much more common, and Dayi has the inconvenience of using up number keys to type, so that the numbers aren't easily accessible. I've only ever used Cangjie, so can't talk about dayi. Cangjie is generally pretty slow. It's got as steep a learning curve as wubihua, but characters take up to 5 keys to type, whereas wubi manages most in 3, at most 4. Also, it's not that tolerant of typos. Shape inputs are becoming less common in daily life.
For those not happy with Cangjie's steep learning curve, there's sucheng（速成）, which only takes 2 stroke inputs from cangjie and tries to figure out the character. But, this involves a really long dropdown list, and while a few people use it, it's really not worth your time if you're trying to type anything quickly (or at all, really). Wubihua only types simplified chinese (usually, though there are ways to), so isn't used at all.
Hong Kong
Hong kong use Cangjie like Taiwan for stroke inputs, with dayi being uncommon, but again not totally unused. Cangjie is probably the most common IME overall in HK. Other common ones are phonetic, though these are far less widespread. The old romanization system of cantonese is used, but also Jyutping, which takes a more modern approach, and is developed by the linguistics society of HK. It excellently matches the sounds of cantonese with the latin alphabet, and the IPA (International phonetic alphabet). Its use isn't widespread, but there are some IMEs which intake phonetics in HK, though these are far less common.
There are more for all, but generally these are the most common.
For linux, compatibility really limits usefulness. Few of the major IMEs are designed for linux, though Google and Sougou have come out with adaptations. If using linux, these are your best bets (though I take it that not many people use Linux -- less than 1% of all computer users, and probably at most 2% of people on Stack Exchange actually use it)
Sources and Statistics:
Dashan on IMEs: http://www.quora.com/What-is-your-preferred-method-to-input-Chinese-on-a-computer-Why
Baike on IMEs (ZH): http://www.baike.com/wiki/%E4%B8%AD%E6%96%87%E8%BE%93%E5%85%A5%E6%B3%95
Sina tech report, probably the most valid to back up my claims (ZH): http://tech.sina.com.cn/mobile/n/2014-05-09/09199368867.shtml
Unable to find any raw data regarding percentages of use in each, though by looking through forums these are generally the most commonly mentioned (with the exception of Dayi, and phone input methods which people generally don't talk about as much).
