Update: this issue also applies to SCOM 2012 R2 as confirmed in this article by Kevin Holman

Microsoft released a KB article on the 5th of December on how to deal with a particular issue in SCOM 2012 SP1. It may be worth applying this fix preemptively if you are still running SP1, in order to avoid unnecessary downtime.

“After an instance of Microsoft SQL Server that hosts the OperationsManager database goes offline, the System Center Management service of the Microsoft System Center 2012 Operations Manager Service Pack 1 (SP1) management server stops responding.

For example, the System Center Management service stops responding after the instance of SQL Server disconnects, restarts, or fails. To recover from this issue after the instance of SQL Server is available again, you must restart the System Center Management service. “

KB Article

NB: As always backup your registry before making any changes

To resolve this issue, you can enable the automatic recovery feature in System Center 2012 Operations Manager SP1. By default, this automatic recovery feature is disabled. 

To enable the automatic recovery feature on the management server, follow these steps:

1. Start Registry Editor.
2. Locate and then click the following registry subkey:
   HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\System Center\2010\Common\DAL
3. Create the following two registry entries:
   • DALInitiateClearPoolType: DWORD
     Decimal value: 1
   • DALInitiateClearPoolSecondsType: DWORD
     Decimal Value: 60Note The DALInitiateClearPoolSeconds setting controls when the management server drops the current connection pool and when the management server tries to reestablish an SQL connection. We recommend that you set this setting to 60 seconds or more to avoid performance issues.
4. Restart the System Center Management service on the management server.