If the wiring is too small for the breaker size, then yes it's dangerous. However, I'd not trust a repairman to know what he is talking about. Have a real electrician, not a repairman, not a contractor who thinks he knows how to do electrical work.

It is not a common thing. I've torn apart every house I've owned and the breakers always are rated for the lowest gauge of wire used. This means often times ppl stick in lots of 15A breakers - but oh well at least it's safe.

If your dryer is on undersized wire that's definitely a problem. It won't get the current draw it needs thru a 12ga wire and if it does it would be a hazard. Of course you could use it this way for years and maybe be fine -- or not. I would definitely repull the right size wire for your dryer, and put 15A breakers on any circuits that use 14ga wire. Switching out the breakers is a cheap fix at $3 ea.

For starters, if you're comfortable, look into it yourself. All your 20A breakers should have 12ga wire coming off them. The gauge of the wire is printed right on the wire. 15A breakers should have 14ga, tho they can have bigger wire.

This won't tell you for certain if 14ga wire wasn't used somewhere you don't see... but it will point out if you have obvious red flags.

I looked at the wires but couldn't see any info printed. For what it's worth, the dryer (a 30A breaker) and a 50A breaker both have huge red wires coming out...much bigger than the other wires connected to other breakers.

I will definitely find a licensed electrician to look at this. Thanks for the advice.

If there is any marking it will often be only on the outer jacket of the cable, not the individual wires in the cable.
#14 is very close to 1/16 inch diameter (0.0641 inches actually), #12 around 1/10 inch (0.0808 inches actually closer to 1/12 inch).

If you see them in the same panel it is readily apparent when they are side by side.