I just shot a few frames of a cloudy sky at f/22, and I have almost identical spots on my image as illustrated by your black and white shot here. I use my smallest aperture very rarely when shooting, and I haven't noticed any issues until now.
I live in Canada, and just got off the phone with the Nikon Canada repair department. The woman I spoke to was extremely kind and deeply apologetic -- she told me the spots were not dust but, indeed, oil spots, and that this is a known problem. She told me to send in my camera, and that they would replace the shutter mechanism as cited by others on this forum. I asked if they could cover my shipping costs, and she sent me a voucher so I can courier the camera to Nikon at their expense. She also said it would take about seven days to get my camera back from the Toronto headquarters.
I will be shipping my camera out this week. I was delighted by their response, and recommend that everyone test their D600s at the smallest aperture to ensure this problem is fixed while your camera is still under warranty. She also noted that not every D600 has this issue. I remain happy with Nikon's customer service, and will be curious to see how this repair fixes things.
PS: My serial number is 5002XXX purchased in October 2012.