Nokia Lumia 820 - How can I silence an incoming call on my Lumia phone?

During an incoming call, before the call is answered, use the volume up and volume down keys to silence your phone without rejecting the call. You can still answer or ignore the call as normal afterwards, as long as the caller does not end the call, or the call has been diverted to voicemail.

From the Lumia Amber update onwards you can silence incoming calls by turning your phone face down. To switch Flip to silence off, go to Settings > Audio > Enhancements and then slide Flip to silence toggle switch to the left.