The comma in these examples indicates an intonation: the part before it is enough for a complete sentence, so the part after the comma can be dropped. So the versions with the comma imply:
(i) She sings.
(ii) He stands.
(iii) He shouted goodbye to his friends.
(i) is okay if it means her habit, and (iii) is okay, but you wouldn't say (ii), so you also wouldn't say (ii) with something non-essential added to it. Without the comma, it's okay because it explains how he habitually stands: He stands singing by the window.
In each example, the participle is a verb and needs a subject. In the first two there is only one person the subject can be, so that's not a problem. In the third, it can be either he or his friends who is running. Without a comma, 'running' would pick up its subject from the nearest noun phrase - 'his friends' - and in fact would be part of this noun phrase. The comma separates these. It allows it to get its subject from something further off.
He shouted goodbye to his friends running towards the house. [his friends are running]
He shouted goodbye to his friends, running towards the house. [he is running - more likely, anyway]