Food Recommendations for Palermo, Ragusa, and Catania Sicily?

I don't recall where I ate, but my only recommendation is to pick places simply by poking your head in and checking them out. Stay off the main/touristy drags and look for mom and pops with Italian menus only. Tourists tend to self-select places with English translations and those that are off main piazzas/streets, so check out the side streets instead. Palermo has a lot of street food you can sample (fried arancini, etc.), and be sure to check out the markets. Also ask your hotel/lodging for recommendations and be sure to say you're not looking for a typical tourist joint.

I really enjoyed Il Mirto e La Rosa in Palermo during my 2015 visit. I had read that it was formerly a vegetarian place (it now has a broader menu) and still had interesting vegetable offerings--something I'm always looking for as a traveler. After my return I checked it out on TripAdvisor and found that not all the reviews were positive, so I suggest that you do a little checking for yourself. It didn't feel like a touristy place, but it's not priced at the super-budget level, so it may not be as "local" a place as you're looking for.

Edit to add: These are all in Palermo.

We enjoyed L' Alcova dei Sapori at Bara All'Olivella 101. Not touristy, great fresh seafood. Not expensive. The owner (Giuseppe) speaks English - he used to live in New Jersey - but the rest of the staff don't.

We also had a nice, inexpensive lunch at "Il Proverbo," just off Via Roma on Discesa del Girdici (? my handwriting is sloppy on that one.) That place was recommended by the staff at our hotel. DH had a great meal from the menú turistico; I had a pasta dish that was good, but a little too salty. I particularly enjoyed "La Provenda," off Via Roma, a few blocks north of the train station. I went there by myself (long story), and had what may have been the best pasta I've ever had - fettucini con funghi.

And don't neglect the street food. We ate several times at the Vuccinía market, again, just off Via Roma. Great fun, and mostly locals there.

In Catania, we ate at a restaurant that had horse on the menu. If you're interested, let me know and I'll look up the name of the place!

In Ragusa, while on the Rick Steves Sicily tour, on our own one night in Ragusa our tour guide Tommaso recommended Quattro Gatti and it was very good, with a pre fix menu for just 18 euros. Run by a husband and wife, very homey.

Here is what I wrote at the time about my trip to Sicily in April-May 2014:

Palermo:
Trattoria da Pino, recommended by our hosts, was a great locals place. Very cheap with a working-class clientele, and lots of fun.
Pizzeria La Duchessa di Citarda Giada has good pizza as well as other dishes. Mostly tourists, but still cheap and good quality.
Sapori Perduti had a lovely and somewhat fancy atmosphere, but the food, while good, was disappointing for the price.
Ciuri Ciuri is a bar ("cafe" to us Americans) with delicious arancini and other local specialties - highly recommended.

Ragusa Ibla:
Cucina e Vino - delicious pastas and wonderful stinco di maiale (pork shank), in a lovely atmosphere. One of the real highlight meals of the trip.