An instruction from ATC to copy a number can mean many things. Whatever the reason, you can be sure that, to ATC, it's significant enough to merit a one-on-one conversation off the radio.
Sometimes, ATC says “possible pilot deviation” before asking you to copy a number. That’s usually a sign that representatives from your local FSDO and beyond will become involved, which is no fun but not necessarily a big deal if you cooperate. Even more likely, a controller just wants to clarify what you were doing during a specific part of your flight or maybe give you a slap on the wrist rather than a gut punch from the FAA.
Regardless, copying a phone number from ATC can cause stress, so the first thing to remember is to fly/taxi the plane because your focus may be thrown off momentarily. Keep things straight and level, and then reach for your pen or ForeFlight scratchpad only when it is truly safe.
Arguing or debating with ATC about why they want you to call is not a good idea. You aren't the only pilot in their workload sphere, and fellow pilots don't want the frequency tied up. The tension that can go out over the airwaves affects everyone. Be professional and help keep the temperature at a reasonable and low level.
Getting into specifics, "Possible airspace incursion" is a fairly common follow-up from ATC. Again, just relax, copy the number, and don't recount the entire history of your flying. All questions and answers can wait until you’re on the ground, out of your aircraft, and able to focus on the task at hand.
Be polite and respectful, and read back the phone number. Then, continue to your destination, park the plane, and take a breath.
Before you dial the number ATC gave you, think through what might have happened to trigger their inquiry. It’s much easier to decide if and how you should have the conversation if you have an idea of what you might have done wrong.
Airspace violations are easy to detect. Stay very aware of the surrounding vertical and lateral limits of airspace, as well as its operating time found in the chart supplement.
Maybe you simply got the tower closing times wrong. In any case, if you have ForeFlight, consider keeping the breadcrumbs on to see where your flight path took you.
Busting a TFR is a reason ATC would request that you call. If you use ForeFlight, ensure you activate TFRs and Airspace Alerts to avoid possible trouble.
Even then, not all TFRs show up on your EFB 100% of the time. Brief well before you take to the sky. ATC expects your situational awareness to be high, so scrutinize your sectional before heading near Bravo, Charlie, or Delta.
But how about some not-so-obvious scenarios that could warrant a number to call? What if you departed a Charlie airport with an airliner on final behind you, and you took just a tiny bit longer to get airborne than ATC was comfortable with?
ATC might wait until you land to give you a number to call because they would have had their hands full at the time. They could just want to inform you that an airliner had to go around because you were on the runway a moment too long and advise you to keep them better informed in the future. Slap on the wrist.
Intermittent transponders can make life interesting, especially if you have flight following and have been confirming your altitude along the way. It doesn't hurt to keep an eye on your altitude reading on transponders to display this information in case a reading is clearly nowhere near what your altimeter says.
Suppose you're on a night flight and see bright lights rapidly climbing in the not-so-far-off distance. You might get the "advise ready to copy" line from ATC and later find out that your transponder showed you being 2000 feet higher than you actually were, forcing a business jet to maneuver around you. You'd probably be instructed to have that transponder looked at right away, which, of course, you would do to avoid future headaches.
How about a flight where your right main tire blows out on touchdown at a Class Charlie airport, causing temporary runway closure? Flat tires aren't typically cause for the FAA to contact you, but if your incident caused a big stink by snarling commercial traffic, don’t be surprised if you get a call asking for details (and how you plan to avoid it in the future).
Whatever the case, do more listening than talking if and when you call, and take notes of the conversation. If follow-up communication is required, you want to have all of that information copied correctly. Plus, the call will be recorded, and the FAA can use the information you provide to decide how to handle the situation.
As mentioned previously, you should have an idea of what you did wrong before you make the call, both so you can prepare for the conversation and decide if you want or need to consult with counsel. If the conversation catches you off guard and starts to sound more serious than anticipated, you should be prepared to end it politely.
Just like when you were in the air, remember that you won’t do yourself any favors by arguing your case. Unless an egregious violation occurred, ATC may decide not to file if the controller can convey the issue and feels that you understand how to resolve it in the present and avoid it in the future. If the controller escalates the situation to the FAA, you don’t want to sound uncooperative in the recording.
The conversation should be brief if you focus on listening, understanding, and refraining from offering unnecessary information. Don’t end up on the phone for more than a few minutes.
Lastly, remember that being given a number to call doesn't necessarily indicate an imminent action against you. It's a request to communicate for clarification and information purposes, with flight safety and increased awareness being the end game.