If we need five common ways to say someone has lost their job, then let's start counting!
1) Unless you have been specifically asked not to, don't forget the original way expressed in the question. To lose their job. This is a matter-of-fact lexical choice, used objectively in either formal or informal settings. It would be a good choice to use when talking directly to the person who lost their job.
John lost his job last week.
2) To be fired. This is a slightly less formal, yet still appropriate lexical choice. It can be used in pretty much the same contexts as number one. It is more commonly used when it is the fault of the employee.
Alice got fired for stealing from work.
3) To be laid off. This lexical choice usually applies to people who lost their job through no fault of their own. Usually the company they worked for needed to save money, and so terminated the employee's position.
Jenny got laid off when Target had to shrink their budget.
The last two I'll let you write out on your own, but I'll give some ideas.
To be handed a/your pink slip
To be let go
To be given the boot
To be sacked
To be dismissed
That's the idea!