Are there any clear-cut fines for flying an aircraft like a Global 5000 which is not single-pilot certified by a single pilot?
And what would be the probability of being caught and how would they catch you?
FAA Order 2150-3B outlines the enforcement actions that the FAA may take. Here are a few:
What exactly the FAA decides to do when it finds out is up to the FAA and subject to the surrounding circumstances. If you moved the airplane from Podunk A 10 miles to Podunk B, or if you decided to buzz the Whitehouse and wave to the President, these would be evaluated differently.
At the very least your license would probably be revoked or suspended, the FAA doesn't take that very lightly (it is not done often). Most likely you would not be given jail time, as this particular violation doesn't fall under one of the criminal violations (see page 4-39 in linked document).
As far as probability, it would be high, as John K says, you need to be around a lot of ground people (fueling, movement, etc). They would catch you by noticing that you are the only pilot in the cockpit when the aircraft starts moving. Not to mention that the aircraft is not single-pilot certified for a reason, it would be difficult to fly it by yourself (not necessarily impossible, but you will probably be noticed by ATC as you try to do a lot more workload than your resources allow).
According to FAA Order 2150-3B (see link in other answer) in Appendix B (penalty guidelines) the recommended penalty for "Failure to comply with operating limitation" is a 30 to 90 day license suspension. That is what I would expect would happen in real life if the FAA investigator thought that the pilot had no good reason for operating without a co-pilot. If there was some good reason and the pilot's record was clean otherwise, they might change it to a warning.
I disagree with the other answers that claim it would be quickly reported. Unless an informed passenger noticed it and reported it, I doubt it would get reported by anyone else. The fuel guys have no idea what operating limitations are or who is operating the plane. In many cases, crew and passengers are long gone by the time the fuel truck pulls up. For the fuel truck to deal with just one guy or to fuel an unoccupied aircraft is commonplace. For all the fuel truck guy knows, there could be crew inside the aircraft doing stuff. He has absolutely no idea what the operating situation of the aircraft is.
The probability of getting caught would be pretty high because you would be around ramp crew all the time and somebody would notice there's only one pilot aboard when there are normally always 2. But you could probably get away with it a few times.
Penalty for that kind of thing would depend on the circumstances and motivations. At minimum a fine of a couple grand and a license suspension for a number of months. A professional pilot who depends on his/her license would be insane to do it unless it was for life and death reasons (like escaping from some dangerous place).
Possibly jail time. Authorities in the 80s and 90s started to revise regulations to add the option of criminal liability for certain violations on top of administrative punishments, to give the rules more bite.