Hello there,
When an inductor saturates it's inductance goes from a relatively high value to a relatively low value, meaning the current rises very sharply and on the scope it looks like a spike of current (using say a current probe). At that point the current is limited by only the series resistance of the coil Rs, so the current would be due to whatever dc voltage is applied divided by that Rs. It can be very very high for power electronics inductors and can blow out transistors unless some means to prevent this is also incorporated into the circuit. Many modern circuits monitor the max inductor current to prevent the power transistor from blowing out, and some even use the inductor max current as a means of regulation.
Oh yeah one more thing worth mentioning is that sometimes the best circuit to use to test an inductor is the actual circuit it is being used in. In other words, check the inductor current in the circuit it is working in presently and see if you can find some mode that makes the inductor saturate. That's the best test.
Just before the inductor 'saturates' the current slope increases, so that's a tell-tale sign that something is going wrong. Many times it will be almost perfectly straight, then curve up near the end of the pulse when something is not quite right.