Getting small pulses on the output of an AND port, even though inputs are always low
If the inputs are truly stable and low, then you are probably seeing spikes on the ground pin of the 7408. It is also possible that you have noise on the power line, but if you only see positive spikes I would suspect the ground connections.
Be sure to have a bypass capacitor directly at the power and ground of all of your logic devices, and distribute power and ground with short fat wires.