I have a basic solid state relay that switches an outlet on and off with a 9v input. Everything works good but I'm confused by the voltage readings I'm getting. The load side is reading 124 volts, and the line side is reading 118 volts, and I'm getting this reading whether the relay is on or off. Is this common?
When in its 'off' state, the SSR will still pass through a small leakage current. You can find its maximum value in the SSR datasheet.
If you have no load on the SSR output except a multimeter or oscilloscope, the very high input impedance of the DMM/scope will draw little current from the SSR. The leakage current will be enough to maintain a high voltage drop across the DMM/scope impedance. Hence the high voltage reading.
You don't say what your SSR load is.
Try connecting a continuous load, like a fan or light bulb, to the SSR output and measure again. You'll find that the small leakage current will be sunk by the load and you'll measure almost no voltage.