For a circuit with a 24bit ADC, a very low noise regulator was required, I used the LT1761ES5-5.
Flicker noise is a problem even at 16 bits, and the "flicker" in the least significant digit is where it gets its name. Usually people do not think of it as being significant when a 40,000 or 200,000 count digital meter has more than 1 count of flicker in the last digit when connected to a clean source, but it indicates excessive noise and a design defect or damage.
For transducers which require excitation, like Wheatstone bridges, the flicker noise can be cancelled out by using the excitation as the reference for the ADC. In practical terms this means that even a noisy regulator like a noisy 7812 can be fine as a reference if it also supplies the excitation.
Low frequency noise is dominated by the reference, and bandgap references are much worse than zener references. Flicker noise from the error amplifier following the reference is insignificant. It is difficult to reduce flicker noise with filtering, so a better way is to use multiple references to average it out. With zener references levels can be low enough not to be a problem in all but the most demanding applications, and I hardly ever see multiple references to reduce noise.