In addition to the romantic, comedic, and horrific suspense, there is the suspense that tragedies generate. Examples such as the prediction of a seer as in "Oedipus Rex" and "Julius Caesar" come quickly to mind. The reader/viewer then wonders if the prediction of this seer will come true.
Suspense is often generated by an author's foreshadowing. In "Moby Dick," for instance, a sinister man steps from behind cargo and tells Ishmael that the ship on which he is going to work has an insane man as a captain. Charles Dickens's "A Tale of Two Cities" opens with the classic passage "It was the best of times, it was the worst of times" that presages conflict and tragic events. Likewise, short stories contain such foreshadowing. Hawthorne's story "The Minister's Black Veil: A Paradigm," for example, suggests a disturbing outcome as the congregation becomes uncomfortable with their minister's having donned a black crepe veil that covers his face.
Certainly suspense has much to do with enlisting the readers'/viewers' interest as they ask themselves, "Will this couple fall in love? Will the main character succeed? Will the main character succumb to fate? What will happen to the victim? etc."
Good question, but just what kind of suspense can vary from story to story.
It would depend on the story (humorous, horror, drama, etc.) and the plot would dictate what kind of suspense, i.e. "Will the hero survive?", "Will the lover reunite?", etc.
Now, as for examples to back up what I am saying, there are several short stories that feature suspense. Jack Finny's "Contents of a Dead Man's Pockets" is great yarn, as is Frank R. Stockton's "The Lady or the Tiger". Saki's works are beautiful examples of nearly every kind, especially "The Interlopers".
Most of those stories involve external stimuli to generate suspense. For short stories involving interal suspense (usually brought on by guilt) I do not feel that one could do any better than Edgar Allan Poe. "The Tell-Tale Heart" is almost perfect in that regard. I hope this helps.
Suspense is usually found is most stories and in short stories as well. One of the most often use of suspense is foretelling. With this we get a hint as to what is about to happen and what can happen in the story which leaves things blank and us wanting more to suspenseful about.