Look I do understand - you have copied the circuits in the data sheet and you have oscillation. So why post a circuit and photo?
The reason is that we would have seen straight away that you had the regulator connected via wires to a lab power supply, and you had non-tantalum electro's.
It turns out you were not actually following the data sheet, and that was your problem the whole time.
The Thompson data sheet says if you don't use tantalum capacitors, then if the regulator is an appreciable distance from the power supply's main filter electrolytic capacitor, you need to increase the input filter cap across the regulator by a minimum of ten times for stability. So 22uF is the absolute minimum, which means 100uF is a comfortable value - you don't want it becoming unstable as the cap ages and gains ESR. A power supply connected by wires is "an appreciable distance" in pcb design terms, and a lab power supply does not always have a large capacitor across its output.
The exact way the input and output caps were connected to the regulator is also critical, and we are still in the dark about that. A wrong way would have been for the two negative leads of the caps to connect in one breadboard row, and then a wire goes from that row to the 7905 common lead. The best way would be to have the regulator plugged into the middle of a breadboard row, with one cap connected on the front side of the regulator, and the other cap connected the rear side.
These points probably would have been mentioned within the first few posts if we had known exactly the test configuration that you had. I was waiting for a schematic as asked for in the third post before replying and no schematic ever came.
Also if we had known that you wanted -5V for an opamp level shifting, we would have said the 7905 was not the right part, and we would have suggested other references, or alternative opamp circuits that could use the 5v as the level shifting reference instead. There is nothing wrong with a 4.7V or 5.6V zener in fact - 100% safe from instability issues. When you leave us guessing, we struggle to help you much.
Richard.