Around a decade ago, the situation was quite different, and there was a tendency towards separation of the two functions. The recent trend, however, points towards combining both functions under one roof. This trend reversal can be observed also outside the euro area.
For instance, in the UK, the FSA was founded in the late 1990s, but recent proposals aim at re-allocating it under the umbrella of the Bank of England. In the US, the FED was also given an extension of its mandate regarding supervision.