Incorrect spelling, explanation: making the present participle or gerund from the verb, the suffix ing is added to the base. The verb is sue and allegedly, sueing might be correct. However, the rule also says that it is required to remove the silent letter e, the same as in the examples: come – coming, hope – hoping, make – making. Therefore, sueing is spelled wrong, the only correct form is suing without the silent e.

Correct spelling, explanation: suing is made from the verb sue after adding the suffix ing to it and removing the vowel e that is not pronounced. In this case, the rule of forming gerund or present participle is applied. The verb sue has its roots in Latin. However, only in Middle English, the form similar to the present one occurred. It was siuen and now it is sue. To sum up, sueing is incorrect because the vowel e is not removed. As a present participle or gerund exclusively suing is correct.

Definition of suing:
verb, appeal formally to a person or organization that has done something illegal,
John is suing her wife for a divorce.
She was suing the city for damage after the fire.

Phrases with suing:
sue for peace – when you want to stop fighting and ask for it officially,
At last, they are suing for peace.