Very easy to do in GPS Action Replay using the "Tracks Toolbox" (button on the top-right of the main window).
Option 1: use the "Filter" button (works great for peaks higher than your top speed)
Option 2: select the pike you want to remove in the (doppler) speed graph, and click "Erase within selection". Click on the spike first, then use the mouse wheel to zoom in/out, and drag to select.
If you click on the spike, that will highlight it in the trackpoints, if you click on that, the "delete trackpoint(s)" button becomes active and you can delete the spike
No worries Sean, it gave me the motivation to get GPSResults working on my laptop, so I could find the best answer.