I came across the sentence
My biggest grievance is grammar mistakes.
I'd be inclined to write it as
My biggest grievance is with grammar mistakes.
Grammar mistakes are my biggest grievance.
What is the correct way to write this, and why?
The verb should be singular or plural based on the subject. In your first and second examples, grievance is your subject, so you use is. In the last example, mistakes is your subject so you want to use are.