My husband gets paid in cash every week. How would he file that on taxes?

The income is STILL reportable as taxable income, irrespective as to HOW he got it.

He needs to report it as self-employment income on Schedule C and Form 1040.

Is he a salaried employee or an independent contractor. If the former, he should get a W2, the latter should get a 1099.

I believe it's a violation of Federal law to not provide a pay statement with each pay. Doesn't matter whether paid by check, cash or direct deposit. If one isn't provided the employer is breaking the law. On the other hand, if the employer is paying him "under the table" and not withholding taxes, then if he tries to file as an employee, he could be getting his employer in serious legal trouble.

I agree that this is a delicate situation.
If I were he, I would straight out ask what the employer is doing about it on his end.
The only reasons to pay cash are not good - usually to avoid having to pay Workmen's Comp and/or be an employer with an EIN and various procedures in place for withholding taxes and paying what employers have to pay.
He could be let go if he asks the wrong way, so be careful.
If the plan is to give him a 1099 at the end of the year, your husband will have to pay taxes on the amount, and it won't be a small amount, because the employer won't cover half.

It appears, this is a problem, since if he is really an employee, the company should be withholding taxes, paying into social security and paying for workers comp insurance.

If they are just handing him cash, with no records, they are not reporting his income to the IRS.

He is responsible to pay all of the taxes, including self employment taxes. ( so he will often owe more taxes)

Normally this is done by small companies, who do not report most of their income to the IRS either, and operate somewhat off the books.