Non inverted OpAmp & Shunt circuit not working
0R1 is too low for the secondary side shunt (RM). If you read the datasheet then you'll notice that all the performance characteristics are given for RM = 10R. You may worry about the dissipation for RM = 10R compared to that with 0R1. So you can use RM = 1R.

Besides, with RM = 0R1 the full-scale voltage will be 25mV for IL = 500A. The offset voltage of the opamp can be as high as 3mV or as low as -3mV, so you may have a measurement error of (±3) x 40 = ±120mV which translates into ±12%. But for RM = 1R the full-scale voltage will be 0.25V, for for 0-1V range a non-inv. amplifier with a gain of 4 (R2 = 3k6 1%, R1 = 1k2 1%) is enough and the error will be significantly reduced.


  I also thought about adding a capacitor of 100nF to the OpAmps PowerSupply, but I think this is not the problem here?


It's always a good practice to place decoupling capacitors right next to the supply pins.