This probably does not even have anything to do with LDO but how you are doing measurement. Say long probe ground lead and noise induced from power source.
and 10uF output
10uF of what type? If it's MLCC, LDO may oscillate.
Going by the data sheet, it should be stable, with a low ESR capacitor. It's not a very low drop-out regulator, because it has an NPN output device. I think the ripple is coming from whatever is connected to the input.
https://www.mouser.co.uk/datasheet/2/389/ld1117-974075.pdfI've seen the noise go up to 500 pk to pk. I am afraid it will be out of the spec for my microcontroller.
I doubt your microcontroller will be damaged, but it could cause problems with analogue circuitry.
The ripple is probably coming from your 12V supply, rather than the linear regulator. You could try an RC filter in the output of the 12V supply. A 1000µF capacitor and 4R7 resistor will have a cut-off frequency of just under 34Hz and will greatly attenuate the ripple which is in the 1MHz region.
EDIT:
Make sure the resistor has a high enough power rating. P = I
2R, so for the full 0.8A, the regulator is capable off I = 0.8
2*4.7 = 3W, but use a much higher rating. I'd recommend at least 8W, as the regulator might not current limit until 1.3A. You could also use three 1R8 3W resistors in series, with a 470µF capacitor to 0V, after each, to spread out the power dissipation and provide a steeper roll-off.