Catacombs in Rome

There are only a few which are open to visitors. and they all require an escorted tour; you can't wander around on your own as some have many miles of tunnels, and some have very valuable artwork and artifacts that must be protected. You will not view human remains as any tombs along the tour route that may been desecrated have had their occupants removed out of respect. These are considered sacred spaces and require proper dress and respectful behavior. Photos are not allowed.

I think few of us have done them all but these three are likely the most visited:
San Sebastiano (I've done this one).
San Callisto/Callixtus (close to San Sebastiano)
Priscilla

https://www.catacombe.org
https://www.catacombesancallisto.it/en/index.php
https://catacombepriscilla.com/en/home-en/

I don't think you can go wrong with any of them. The "bone crypt" of the Capuchin Friars comes up on some itineraries but it's not a catacomb, and is many centuries newer (17th century) than the 3 sites listed above. It's where you go if you want to see human remains but the history of burial practices of the roughly 2nd-5th centuries is much more interesting, IMHO.

https://museoecriptacappuccini.it/en/

I encourage people to visit the Catacombs of Priscilla. I used to live near there and visited with groups many times. Why these?
1. Never crowded
2. Always had a great tour (in English)
3. Important Early Christian art
4. Easier to access than the catacombs near/on the Appian Way (in my opinion)

I am with Kathy. I don't think it makes much difference. Once underground, they all look the same. I would visit on Sunday when the traffic on the Appian Way is restricted to buses and taxis.

Are you interested in even a short (for example 30 min. walk) on the Appian Way? If so, that would be a factor in choosing to visit the catacombs near the Appian Way.

Are the catacombs free on the first Sunday like the colosseum? My wife and I are walking the Appian Way on March 2nd.

No, free first Sundays only apply to Italian State Museums. The list is here:
https://www.turismoroma.it/en/events/first-sunday-month-free-entrance-state-museums