The manual can be found at
https://www.singer.com/support . The needle needs to be inserted with the
flat on the right. Does you machine have a thread guide under the needle clamp? Have you made sure the clamp screw is long enough and not broken off? I think that the suggestion to take a mirror and see if the screw enters the hole where the needle goes to make sure it is long enough to actually clamp the needle. Or I suppose you could unscrew it completely and take the whole clamp comes off. I'm not sure how the thread guide fits in the needle clamp. Perhaps someone didn't get it in correctly and that is why the needle falls out.
Janey - Neat people never make the exciting discoveries I do.