Yes because Mobile phones don't use sound waves to communicate. They use a variation on radio waves that do in fact travel at the speed of light (in a vaccum that is) so yes they could in theory work

but

No because the key word isn't "mobile" is cellular. This is important because it refers to the footprint of the towers that are in cells like honeycomb. And since because a plane flying supersonic is likely to be shifting from cell to cell rather quickly it seems to me to be rather unlikely that you could use a standard mobile phone but if you wanted to give your character a sat phone like the ones marketed by the now defunct iridium* sat phone service it could work.

*Historical Aside: Iridium tried to get off the ground in the mid to late 90's when cell phones were just catching on. They had plans for a total constellation of 33 satellites to prvoide global coverage. I think they only got something in the neighborhood of 8 or so up before going under but i could be wrong.

Howevever, a "cellphone" tied into the aircraft's own communications equipment can be used, so you can call a passenger in-flight or a passenger could call you (this service typically costs extra).

As for whether a conventional "ground" cellphone can work in-flight, the answer is "sometimes". When flying over an area where coverage is provided by many closely packed cells, the hand-off's between various towers would increase the chances of a dropped signal dramatically (compared to using it in your car or something). On the other hand, if you were flying over an area with larger cells (like a rural area), it wouldn't be such a big problem. The fact that you're at a different altitude than the system is designed to accomodate and you're inside of a large metal container that has its own electronics will also have an impact.

So I would think it could be done in the other direction (calling from ground to plane).

If you're using your cell phone in a moving plane (and also cars), you'll only stay in touch as you're in range of the particular cell phone tower that's picking it up. On 9 / 11, when Barbara Olsen called her husband Ted, she called and they talked, then they were cut off when the plane moved out of range, then she called again and they talked until the plane again moved out of range, a couple of times until Ted Olsen didn't hear from her again.

The banning of passenger use of personal communications devices and most other electronic equipment derives from worries about interfering with the plane's own electronic equipment. Most of this worry dates from a plane crash in the late fifties or early sixties, where an early commercial jet was thought to have been brought down by a portable shortwave set someone brought on board. (Only they were never able to duplicate the effect in the lab.) Cell phones were used by the plane passengers on 9 / 11 because almost everybody has them now, and such was the stress of the incident that those particular rules were junked, if the passengers even knew about them.

I thought planes that flew faster than the speed of sound only made sonic booms going up past or coming down below that speed, not when they were actually above it.

Edited to add: And they are continual as the object moves along at supersonic speeds. They may seem to be short in duration because they have moved past the hearer.

http://www.sky-flash.com/boom.htm

If you call someone on their personal mobile phone, whether it's a sat-phone or whatnot, then they won't be allowed to talk because the steward/esse/s will confiscate the phone in question. They're pretty lax sometimes about allowing people to turn their phones on in-flight (which is technically just as prohibited), but they definitely will take your phone away if you start yakking.

By the way, I had the privilege of being at the Cape when a shuttle landed. Didn't look fast enough in the right direction to see it, but I heard the sonic booms (yes, plural--one boom! quickly--as in a fraction of a second later--followed by another).

I have asked around about why there would have been two sonic booms, and the best we can come up with is that maybe the tip of the nose generated the first boom and the front edge of the tailfin generated the second one.

But, I remember on a couple of the sites it did say that sometimes you could get a double boom. One from the point of the plane & then one from any other bit of the aircraft that sticks out more prominantly & causes a second pressure wave, outside the radius of that created by the nose.

So, yeah - you can have a double one.

Incedently - the guy who is receiving the call in the aircraft is okay to receive it. It's outside of normal passenger jet circumstances.

In Tom Wolfe's account of Chuck Yeager's first supersonic flight in "The Right Stuff," I believe it mentions the double boom, and that it was in some way a predicted phenomenon, but didn't as I recall go into clinical detail. (I say "I believe" and "as I recall" because my copy of "The Right Stuff" is inaccessible behind a big pile of boxes in my so-called library right now, so I can't check it.)

I've never seen a shuttle launch or landing up close, somewhat to my regret. Though I've never spotted a landing from where I usually am in Florida, if launch weather is particularly clear, I can see the trail left behind as it rises, all the way up past booster separation. (I could see the recent Pluto launch, as well.)