Had the same problem => Plugin QMYSQL & QMYSQL 3 were available but couldn't load them :
"QSqlDatabase: QMYSQL driver not loaded / QSqlDatabase: available drivers: QSQLITE QMYSQL QMYSQL3 QPSQL QPSQL7"

Works for me on Windows with Qt SDK 5.6.0 (32bit version with mingw 4.9.2 32bit included) :
- Downloaded MySQL 32bit in zip archive (mine was v5.7.11)
- Extracted it to C:/MySQL
- Copied "C:\Qt\QMySQL\lib\libmysql.dll" and "C:\Qt\QMySQL\lib\libmysqld.dll" to Windows/System32 (to avoid the "cannot found -llibmysql" compilation error)
- Compile with "How to Build the QMYSQL Plugin on Windows" on http://doc.qt.io/qt-5/sql-driver.html#qmysql (my command was "qmake "INCLUDEPATH+=C:\Qt\QMySQL\include" "LIBS+=C:\Qt\QMySQL\lib\libmysql.lib" sql.pro" and "mingw32-make")
- After compilation succeeded, copy (the freshly compiled) Qt5Sql.dll & Qt5Sqld.dll from "C:\Qt\Qt5.6.0\5.6\Src\qtbase\lib" to your application build directory
- Restart QtCreator, that should work