how did you settle on 10uF as the optimal choice for this particular LDO especially since a tantalum is recommended?
Never said that's the perfect choice. Mine was acting strange, searched online for that model of supply, and found out it's a known issue for AMS1117. The 10uF electrolytic was what it happened to be at hand. Used them as a test, and the oscillations went away, and that was it.
Searching what would have worked even better would have been a waste of time. Particularly for an item that was virtually free (a set of 65 breadboard wires + 830 points breadboard + 3.3/5V supply + shipping included from Aliexpress was $5.78 total).
I'm not very sure why it is so important for you to use the "optimal" value. Striving for achieving the best, or striving for excellence as a life guideline is laudable, but in practice you'll have to stop optimizing once a goal is fulfilled.
could you please recommend any good reads on the phase margin method?
That AN note was one. But that makes sense only if your background were EE, or Mechatronics, or something else where the control theory is part of the curricula.
There are many free classes on MIT OpenCourseWare, or on Khan Academy, all very good. From the top of my head, I remember a YT channel I was following at some point, with a guy that also published a free e-book about control theory in case you prefer written instead of video, has video playlists, too, as if it were a semester or two of classes:
https://www.youtube.com/@BrianBDouglasMy advice would be to leave these for later. Saying so because, by the original post here, I'll guess you stepped into electronics recently, and most probably this is your first oscilloscope, which is absolutely great, there is so much to explore. Maybe you are getting ahead of yourself jumping straight to control theory.
Unless you already have the prerequisites for it, won't make much sense, it's an advanced topic. The EE prerequisites would be AC circuits, Laplace and Bode plots, maybe some other topics in between about feedback, filters, and the effect of poles and zeroes. All these take 1-2 years to learn in an EE university.
Once you become familiar with that, it's just a matter of plotting the closed loop response against the open loop response, and measuring the angle at which the two lines intersect with each other.
There really was not any noticeable fluctuations
In this case, looks like your breadboard supply was not oscillating.

If the oscilloscope you are using does not have enough bandwidth are there techniques to ascertain if oscillations exist without resorting to other equipment?
One way could be to estimate the expected temperature of the chip, then to measure the real temperature. If the temperature is too high, it means some oscillation is heating the chip. The temperature can be estimated by the thermal resistance and the quiescent current, usually given in the datasheet. The input voltage is known (9V), so the dissipated power and the thermal resistance will give the expected temperature.
I messed around with this some more and I do believe I figure out what the issue is. The short answer is it is the 9 V 1 A wall wart.
Well done, congrats!

Measuring the wall wart WITHOUT the ground clamp ( i.e. removed from the probe) I got 167 Vpp at 60HZ. This is pretty much exactly half of the mains Vpp.
That's an interesting observation, but I guess it's only a coincidence. With only the tip of the probe attached, it would be expected the observed voltage to vary wildly with the surroundings. For example, it is expected the Vpp will vary a lot by simply putting your hand around the wire or around the wall-adapter (isolated, without even touching). It changes because the parasitic capacitance between the body and the wire. Even if you just lay down the probe on the table, and come with the hand near the parrot clip, it is expected to see a lot of induced voltage from your body.
Without the ground wire, the probe acts as an antenna.