"My stomach hurts." The reason is very hard to explain. In present continuous ("I am talking," etc.) we don't like to use certain verbs, such as like, know, have, want. For example, "I want a burger" is correct and "I am wanting a hamburger" is wrong. These verbs -- like, know, have, want, etc. -- are called "stative verbs." Stative verbs never go with present continuous. However, it becomes complicated here. Some verbs have a regular "active" meaning AND a different "stative" meaning. For example, the verb "have": I HAVE two dogs. (stative) What time do you usually HAVE breakfast? (active) If the meaning is active, you can use the present continuous. So "I am having two dogs" is WRONG but "I am HAVING breakfast now" is RIGHT. "Hurt" is just like "have." There is an active meaning and a stative meaning. My stomach hurts. (stative; there is a pain in my stomach) Stop it! You're hurting me! (active; your action is causing me pain) I hope this helps!

They are both right but most people tend to use the first one more often.

You can use both: "My stomach hurts all the time." "My stomach is hurting badly now."

well, actually i'm not a native speaker. But i'll try to answer your question. I think the right is the 1st sentence.

neither of them is right it can be said my stomach aches or my stomach's aching