Most employers intentionally have foreign employees on part time, and thats their excuse for not getting a work permit. I can tell you hundreds of people in that situation. That is the reality.
The majority of employers do not want to deal with work permits, etc. That's a reality. And that's established places like universities, or big film production companies.
Yes, what is the 'right' visa. Some are evading the catch-22 here.
I know numerous teachers who are constantly dealing with visas, as the only places that give a work permit are international schools.
Universities do for full time, but I know many teachers at uni's who are put intentionally on part time, so they don't have to deal with the red tape of getting them a work permit/visa.
None of the English schools that I know offer that.
I know many in the film industry. It's even worse. As work is project to project/contract to contract. So if you're doing a commercial, you do sign a contract, that is in a sense a temporary work permit for the amount of time i.e. 1 day, 1 week, 1 month. Same with working on films, it's just for that period.
So some of the answers /responses on here aren't dealing with reality of those that aren't married,nor in a corporate work permit position.