I think the fact that it was a video allows the singing to be considered either way "” as a completed action (gerund) or as an action in progress (participle).
A lot depends on the nature of the verb phrase. If the -ing form is perceived as a completed action it is a gerund and can be used, in formal English, with the possessive. If, on the other hand, it is perceived as depicting an action in progress, it is a participle.
Not all verbs allow the possessive + gerund. For example,
"” I saw a video clip of you laughing (not *your laughing)
"”I saw a video clip of you playing (not *your playing) with the new puppy
If it were a still photo instead of a video, the possessive would not be used at all. The -ing form would be a participle "” an action in progress.
"” I saw a photo of you singing... (not *your singing)
"” I saw a photo of you screaming at age three (not *your screaming)
"” I found an old picture of my mother cooking (not *my mother's cooking)
Likewise, an aural medium would yield
"” They discovered an audio tape of the artist talking about his work (not *the artist's' talking...)