Correct spelling, explanation: suing is a present participle derived from the verb sue. The rule says that we should add the ending ing to the base after removing the silent e that is not pronounced. As a result, it is spelled as suing not sueing. The word comes from Latin as sequere, a completely different form but in Anglo-French, it occurred in something similar to the present form. It was suer and in Middle English siuen. Now the form is sue but as a present participle or gerund, the correct form is suing, not sueing.

Definition of suing:
verb, if you sue someone, take legal action in court against him in order to seek justice,
John is suing the company for money because of his poor working conditions.
The article smeared her private life so she decided to sue the newspaper.

Phrases with suing:
sue the pants off (of) someone – take legal action against someone for a lot of money,
She was suing the pants off of them after the car accident.
sue out (something) – to take out some written legal order,
The policeman is suing out a search warrant for his house.

Incorrect spelling, explanation: users of English know the rule that forming the present participle form the verb sue they should add the ending -ing. However, it is not enough to make the correct form. Because the verb sue has the non-syllabic vowel e this unpronounced letter should be removed. It is very important to remember that. To sum up, only suing is correct, sueing is spelled wrong.