Have there been any solo flights on airliners?
In normal operations there are regulations against this sort of thing (as Dave has said).  But there have been a few emergencies where the First Office or the Captain have become incapacitated (due to medical emergencies), and the remaining pilot has declined to ask if there is another pilot on board to help fly the plane.

Here is an example from 2011 of a Ryanair flight having this occur**, I have heard of others but can't find references.  So, admittedly, that may be the only actual occurrence.

So, to finally answer your questions:  Very rarely, in an emergency situation, there can be a single pilot on a commercial airliner.  Granted, there were passengers on board, so I'm not sure it fits your narrow definition of "solo", but it's about as close as it's going to get.



For completeness (though I don't think it counts), there have been a few instances of a single pilot stealing a commercial jetliner.

For example this 727.  There were actually two people involved, one was a mechanic, the other had a private pilots license (hence the pilot was "soloing").  The aircraft has not been seen since...so it's assumed it ended badly, and, for the record, it's also assumed the pilot was coerced (if indeed he was even flying the plane).

There are a couple more incidents, namely this Antonov-26, which was stolen by a mechanic, circled for an hour, then crashed.  And also this ATR-42-320, which was actually stolen by a Captain form Air Botswana.  He circled the airport for a couple of hours demanding to speak to everyone from his girlfriend, to his boss to the President of Botswana.  They were actually arranging to have him speak with the president when he crashed his plane into two other Air Botswana ATR-42-320s that were parked on the ground.  He was the only one killed.

Again, though, I'm not sure if these fit the criteria you are looking for.  But in that last one, at very least, he was the only person on board the aircraft, it was a commercial aircraft and he was in fact a Captain.

But to say all this an irregularity would be an extreme understatement...



** For clarification:  The reason I state this is the only example I could find is because it's the only one where I could find a report where it was specifically stated the remaining pilot had no help.  To quote from the report:


  Once the initial Cabin Crew assistance was complete, the Captain indicated that he did not need assistance in operating the aircraft, although this was suggested in the operator’s procedures for pilot incapacitation. The First Officer took no further part in the operation of the aircraft. No inquiries as to the possible presence on board the aircraft of either medically trained personnel or qualified pilots were made.


All other examples I saw either specifically stated that a new co-pilot was found, or they skipped that detail entirely.

All of that being said, I somehow doubt this is the only case, it's just the only one I could explicitly verify.