This message started off as a request for help, but at the end I found the solution, so I am still sending it in case others have similar problems:
I am using a UHFSDR with SoundBlaster Live USB unit, and a QRP2000 USB controller for the Si570. The PC is an old Medion Pentium4 2.6GHz running windows XP. The QRP2000 is in the same diecast box as the UHFSDR, but fairly well removed from the audio amps and RF input. My problem is that around the centre spike of the spectrum I get spikes at exactly 1kHz intervals, out to a distance of +/- 10 kHz. The two nearest the centreline are -110dB and they tail away to the noise floor at -140dB. If I disconnect the USB cable feeding the QRP2000 the spikes are still present but 10dB lower. Disconnecting the QRP2000 from the UHFSDR ADUM1251 makes no difference. I can sometimes see them by touching the audio input to the Soundblaster card, and turning off the UHFSDR 12V supply makes little difference.
Solution - found that connecting the line input to the SoundBlaster card to any ground forming a ground loop picked up the spikes worse, so they weren't coming from the UHFSDR.
Eventually, found that putting 6 turns of the USB lead feeding the Soundblaster Live unit through a toroidal core cured most of the problem. Small cores did nothing, but the core of a toroidal mains transformer worked well. In the end I used the ferrite core of an old line output transformer. The interference was reduced to just two spikes at 1kHz from the centre line at -120dB. I presume that the switched mode PSU in the PC was the culprit.