Should I expect my power line to be this noisy?
Your schematic shows no decoupling capacitor across the supply.  High-current spikes, like the 555 generates on transitions, require a supply that is low-impedance at high-frequencies.  The inductance and resistance of the cable connecting to your supply is too high for it to supply those high-frequency currents.

Try adding a capacitor of a hundred or so microfarads across the supply, as near to the 555 power pins as you can get.

Your scope shot is shows noise with a period of 3.5 * 200 nS, or about 1.4 MHz.  This is a higher frequency than any noise generated by the 555 circuit.  It could be residual switching noise from your power supply, but is most likely sloppy scope probing.  If you are using the alligator-clip-on-a-wire ground lead on the scope probe, you have made a little loop antenna which will show you any magnetic field in the vicinity.

To see how the 555 operation is affecting the power you need to be triggering synchronous to it.  Connect one channel to the LED and trigger off of it, then look at the supply with the other channel.