You can determine if your backup is complete by looking at the Status tab, which gives you an estimate of how much of your backup has completed and what remains.
The "Overview" tab shows how many items you have already uploaded and how many remain.
The "Queue" tab shows you what files are in line to be encrypted and uploaded to the SpiderOak servers and how much progress has been made on each.
The "Actions" tab lists all actions that SpiderOak is taking: uploading, scanning, syncing, etc.
The "Log" tab shows you which files have been uploaded to SpiderOak.
Note that the information on this tab resets each time you close SpiderOak, so it only displays files which were uploaded since the last time you opened SpiderOak.