In “can hear singing”, is “singing” a verb or a gerund?

Solution 1:

Singing is a present participle in the case (following @tchrist ’s suggestion) of “I hear children singing.”

It would be a gerund, i.e., serving in lieu of a noun as direct object of verb hear, if you got rid of the children: “I hear singing.” But in “I hear children singing,” the direct object of the verb is children, and singing is an adjectival modifier thereof, a common function for participles.

A partially analogous construction is available using the past participle, as in “I heard the aria sung,” though the use of the past participle makes the construction passive where the present makes it active: the child sings, the aria is sung.