Partial answer, here is one notable example.
Why does X-shooter use double passes through prisms for Echelle cross-dispersion instead of gratings? informs us that prisms are heavily used in this amazing instruments. (related: What are the pros and cons of different types of echelle spectrograph cross-dispersers?)
- Are there any notable cases or observations where this is/was very important?
The concept of X-shooter has been defined with one principle goal in mind: the highest possible throughput over the wavelength range from the atmospheric cutoff to the near infrared at a resolution where the instrument is sky limited in a half hour exposure.
says X-shooter, the new wide band intermediate resolution spectrograph at the ESO Very Large Telescope.
The option to use a single slit in the telescope focal plane was rejected because of the difficulty of designing a highly efficient relay system and atmospheric dispersion correction for the full spectral range...
The optical design allows the introduction of two short-wavelength atmospheric dispersion correctors (ADC) and the focusing of the target on the slit units at the entrance of the respective arms.
Interestingly as the telescope tracks near one wavelength **it is necessary to also correct tracking for atmospheric dispersion as well:
For slit observations (but not IFU) these tip-tilt mirrors also compensate for shifts due to atmospheric differential refraction between the telescope tracking wavelength (fixed at 470 nm) and the undeviated wavelength of the two atmospheric dispersion correctors (for UVB and VIS arms, see Sect. 2.2.7) and the middle of the atmospheric dispersion range for the NIR arm.
Here is the bulk of the description:
2.2.7. The focal reducer and atmospheric dispersion correctors
Both UVB and VIS pre-slit arms contain a focal reducer and an atmospheric dispersion corrector (ADC). These focal reducer-ADCs consist of two doublets cemented onto two counter rotating double prisms. The focal reducers bring the focal ratio from f/13.41 to f/6.5 and provide a measured plate scale at the entrance slit of the spectrographs of 3.91′′/mm in the UVB and 3.82′′/mm in the VIS. The ADCs compensate for atmospheric dispersion in order to minimize slit losses and allow orienting the slit to any position angle on the sky up to a zenith distance of 60°. The zero deviation wavelengths are 405 and 633 nm for the UVB and the VIS ADCs, respectively. During slit observations, their positions are updated every 60 s based on information taken from the telescope database.
Since the IFU comes ahead of the ADCs in the optical train, no correction for atmospheric dispersion is available for IFU observations, and the ADCs are set to their neutral position in this observing mode.
The NIR arm is not equipped with an ADC. The NIR arm tip-tilt mirror compensates for atmospheric refraction between the telescope tracking wavelength (470 nm) and 1310 nm which corresponds to the middle of the atmospheric dispersion range for the NIR arm. This means that this wavelength is kept at the center of the NIR slit. At a zenith distance of 60° the length of the spectrum dispersed by the atmosphere is 0.35′′, so the extremes of the spectrum can be displaced with respect to the center of the slit by up to 0.175′′.
I can't find a separate image of the dispersion correction prism, probably because it is placed before the entrance slits of the spectrometers.