'There seem' or 'there seems' + usage of the word 'seem'

First, I have a question "How words 'seem' and 'there' are used together?"

Which is correct:

• There seem ...

or

• There seems ...

Then, I'm am interested in general constructions with the word 'seem'.

What are common mistakes people make when they use this word?

Singular:

There is a problem with the car.
A problem seems to be afflicting the car.
There seems to be a problem afflicting the car.
It seems to be a minor problem.

Plural:

There are multiple problems with the car.
Multiple problems seem to be affecting the car.
There seem to be multiple problems affecting the car.
They seem to be minor problems.

Do not confuse with:

She seems to have problems with her car.

vs

They seem to have a problem with their car.