There are two major distributions of MinGW that I know of.
1. The Official Build from the MinGW team, http://tinyurl.com/65ao33u
2. TDM-MinGW from Twilight Dragon Media, http://tdm-gcc.tdragon.net/
(other than versions that come bundled with IDEs like Code::Blocks, CodeLite, wxDev-cpp etc...)
Personally, I like the TDM version because it's easier to install the 64bit version, and statically links to libstdc++ by default so I don't have to worry about sending executables to friends ;)
I was led to believe that libstdc++ is implicitly linked by gcc by default. The official build seems to produce a portable executable out of the box. I'm not sure about the 64-bit version, my computer is still behind the times a bit, so 32-bit is still here with me :/.
I have, however, had problems with TDM-MinGW. For one, everytime I try and setup an environment, I get plenty of strange STL errors. When I try and fix them, it gets so confusing I usually give up and go back to my default environment. I haven't tried it for about a year now though, so I might try again since the official build is pulling some fail sauce on me right now T.T.
EDIT: TDM has the same problem. To be expected actually...
Just setup TDM. Although it seems to not bring much to the table (even crashes in the same place) the website is a lot more straight forward than the official. The installation is also a tad easier and straight forward. There's even *gasp* version control.
It's implicity linked, but it uses a shared library. An executable compiled with vanilla MinGW will fail to start on a different machine that doesn't have MinGW installed.
There's a compiler switch to tell it to use the static library instead, TDM does the opposite and uses the static library by default (which I prefer).
It's funny that you mentionned getting STL errors using TDM, because I was working on some stuff that used some real Template-Fu and was getting odd errors from STL headers, but compiled fine with MSVC++. I tried vanilla MinGW but got the exact same errors. Bug report time!