Let me talk a bit about the difference between NOISE and an OSCILLATION, in particular how it relates to a three terminal regulator suspected of being unstable.
Noise is a property of most physical processes. It is a result of the random motions of the atomic particles that make up matter (and a few other processes) and we don't normally perceive it because we look at things at too large a scale. However, if you look close enough, you will usually find a noise component in any measurement. Electrical noise in the input circuits of your TV set results in the "snow" or fuzz you see when the TV is not tuned to a valid channel, and is also the hiss you hear on your FM radio between stations. It is RANDOM signal without a particular period or amplitude and in fact it's amplitude can only be meaningfully measured when averaged over time.
This noise of course doesn't do any good when present on the power supplies of a sensitive receiver that is trying to find signals already on the order of a microvolt. The noise leaks into the various sections of the receiver and pollutes the desired signals. So we employed a low-noise regulator to not only regulate the power supply voltage at a nice, even 5.0 VDC but also to not add any noise of it's own. Most engineers are not aware that 3 terminal regulators like the 7805 are pretty noisy on their output - they hook up their scope to the output, crank up the sensitivity, see nothing but a flat line - and assume there is no noise. Wrong. The scope just isn't sensitive enough to see the level coming from the regulator. The MIC5209 isn't perfect either, but it's a lot better than a typical regulator, mostly because it has an added pin where you can bypass the feedback loop and lower the noise gain of the part. You can do a lot better with a discrete design, but that's another story.
This is the noise the Micrel data sheet is talking about in the plots labeled Noise Performance. Note that the measurement is in microvolts per root hertz, so you need to know your measurement bandwidth before you can calculate total RMS noise voltage expected.
We don't have a NOISE problem with this part.
We do have an OSCILLATION problem with this part.
An OSCILLATION is a periodic variation in voltage caused by an instability in an active (not passive) electric circuit. Sometimes we want an oscillation as in the voltage controlled oscillators and crystal oscillator, but an oscillation can potentially occur in any circuit with gain or amplification. In a intentional oscillator, the frequency is controlled by some sort of resonator that is in a positive feedback path of an amplifier. The amplifier boosts the amplitude of an initial signal with each pass around the amplifier and feedback path, and the resonator keeps the signal on frequency. The result is a large signal output on the resonator frequency.
However, many circuits use amplifiers and feedback, and are never intended to oscillate. The MIC5209 is a good example with an error amplifier measuring the difference between the desired and actual regulator output voltage and then controlling the pass transistor in the circuit to minimize this difference. The circuit is intended to only have negative feedback around the amplifier which should be a unconditionally stable circuit. But the interaction of the regulator circuit with external loads and impedances (like the 10uF cap in question) can cause excessive phase shifts around the feedback loop and suddenly negative feedback turn positive and we have OSCILLATION. The trick to avoiding oscillation is to avoid presenting these impedances to the part, and that's what Micrel is trying to tell you with their suggested values and minimum equivalent series resistances (ESR). It's tough to come up with these values on your own because it requires simulating the guts of the IC, something we can only guess at the particulars of.
This is not noise because it's periodic, not random. If you look at it on a scope, it looks like a regular sine wave at around 600 kHz. If it was noise, it would look like fuzz on the trace because it's not periodic and the scope cannot synchronize to it.
Conventional wisdom might also say to reduce this signal, add more capacitance to attenuate the AC signal by shunting to ground. Now it's easy to see that adding more capacitance might actually compound the problem by introducing more phase shift.