grammar - Singular or plural verb after "what"

This is an example of a free relative clause. The relative pronoun can be singular or plural, depending on the context.

In a bound relative clause, the relative pronoun has the same grammatical number as its antecedent, but a free relative clause has no antecedent. When you're not sure which number to use for a free clause, you can often deduce it by comparing an equivalent independent clause with a demonstrative subject. For example:

These look like trumped-up charges.
This looks like trumped-up charges.

Both of these sentences are grammatical, with slightly different meanings. In the first sentence, the subject refers to the charges; in the second, it refers to the situation:

This looks like [a case of] trumped-up charges.

Many writers will prefer the first example because the grammatical number obviously agrees throughout, but both are correct. Likewise, both of your examples are correct, although many writers will prefer the first because of the consistent number:

He is being tried on what look like trumped-up charges.
He is being tried on what looks like trumped-up charges.