Which IMEs (input method editors) are actually most used in each of PRC, ROC, and Hong Kong? Because there are many kinds of input method and many implementations of each kind of input method, it's not really practical to learn many.

PRC, mainland China uses simplified Chinese characters, as does Singapore. I assume most people in these places would use a Pinyin IME. But it would be good to know which Pinyin IME is most used currently. If I choose to learn the most popular one I will have the best chance of finding somebody who can help me when I have questions about it.

Hong Kong and Taiwan both use traditional Chinese characters.

Here in Taiwan most people do not use a Pinyin based IME, there are several IMEs based on bopomofo and others based on character shapes. Do we know which type is most used? And do we know which specific ones of the main types are used?

I don't think Hong Kong uses bopomofo and with their changing demographics from Cantonese where pinyin wouldn't work to Mandarin, I'd be very interested to know which kind of IME is most used there, and which actual implementation.

Best answers will be able to refer to some official website or statistics. For people who haven't travelled to these countries and who are just trying to learn Chinese, the IMEs from Chinese and Taiwanese companies and ones that can be downloaded from websites that don't have any English are probably not well known. People learning Chinese with a western language background are probably most likely to use IMEs that come with their computer or OS.

But it would be very good for us to know which ones actual Chinese speakers really use.

One issue though might be lack of English menus, dialogs, settings, and help for the IMEs that don't come with your OS. Learners are not proficient enough to use those in Chinese.
Many people in Hong Kong use Quick aka 速成 or Simplified Cangjie.
There is a wiki link for this input method:Simplified Cangjie

There is a build-in Quick IME in Windows and Mac. Most of the Quick users use it.

Quick users type Chinese using Quick on smartphone too, as the build-in IME of smartphone that selling in Hong Kong usually support Quick. 

However, some people prefer to type using 5-stroke input method(5筆畫輸入法) on phone. There is a wiki link for this input method:Wubihua method. This is a very simple input method. It is based on the stroke order of a Chinese character so as long as you write Chinese, you will know how to type.