From experience, most airlines allow passengers to bring a personal item on-board in addition to a carry-on item. I nearly always travel with both but once while flying Amazonas in Bolivia, it was not the case and I had to scramble stuffing things from my carry-on into my personal item and the rest into a piece of checked-luggage. Luckily I had spare space but this is rarely the case.
For information on what is a personal item we already have this question. We already have plenty of separate questions about specific airlines allowing personal items or not: Spirit, EasyJet and Hop for instance. Plus, I recently had to ask the same of Gol since it's another carrier which I have not flown with before.
Which airlines do not allow a personal item in addition to a carry-on??
Could we have a list for all such airlines, since they appear to be the exception rather than the norm, in order to know which airlines to avoid when needing to carry a personal item as well as a carry-on?