Discontinuity in the charging of an inductor
How are you measuring the current ? A sense resistor, or a current clamp ?

Small amounts of capacitive coupling to your current sense R, or poor common-mode rejection in your scope will cause this type of waveform.

Also, if the inductor is relatively large (physically), it is possible there is some capacitance between the ends of the windings -- you don't have a pure inductance -- and so the total current can have discontinuities.

(I see your schematic now).

Check how your scopes' probe is grounded -- it should be just at the R also. In addition, sometimes scopes can pickup common-mode signals -- try to connect both the probe and the ground clip together to the bottom of the R -- you should see no signal, but anything you do see is in indication of this.

What diode are you using -- if it is not a fast one, reverse recovery current will actually cause large transient currents and noise spikes in the circuit.