This answer is for people like you or me who don't really do this kind of thing intuitively very well and are more "solution-oriented" as you call it...
How should one enquire and be 'emotionally supportive' in this kind of situation ?
If you are talking on the phone or face-to-face, I would try to keep your side of the conversation short. It is not about the actual words, but about your sincerety. For example, when you are in the same office room and they tell you that their parent just died three days ago, there is very little you can really say. Obviously they are getting along, or they would not be here. No, you can't do much to help, in practical terms, anyways. But you can give them a very short and heartfelt condolence, with some body language (i.e., turning towards them, giving them your fullest attention, making a deliberate pause after whatever you said to give them a way to say something, and so on).
Generally, I try to not start to tell similar stories (i.e., no "oh, I remember how it was when my parent died 2 years ago"), and really skip factual stuff. If you know them well, then a "how are you holding together" maybe, but that would be about it.
If they start talking, then of course give them as much space as they need. If they shut down, then don't try to force them to talk.
I'm looking for the medium (phone call, SMS, IRL),
I'd pick SMS/mail/messenger or IRL depending on whether IRL is possible; phone call last. Unless they are very close (close friend/family), then phone call instead of SMS etc.
what to say (point out the positive aspects, ask the person how she feels, etc.),
I'd skip pointing out positive aspects. Unless you really know what you are doing, such things might drive someone who is borderline overwhelmed over the brink as much as help them. Asking them how they feel... well... I'd skip that as well, it would feel hollow to me ("my parent just died, how do you think I feel?").
If the situation is still "hot", tell them some kind of condolence, and if they don't particularly respond, leave it at that. What that is depends a lot on your relationship... between close friends, a "dang, that sucks" might be appropriate; not so much between coworkers etc.
and when to say it (right when I know it, a few hours after, days).
If you have SMS/mail/messenger, then right away. They will read/respond when they're ready.
If you have to call them or walk up to them, then maybe rather give it a day. Obviously, if you suspect that they actually need practical help, then don't hesitate.
Be sure to follow-up after a few more days, a week, 2 weeks; just a small check. And try to find out from their reactions if they need more than they let on. If so, then don't bother asking them if they need help, but offer help instead. Not "let me know if I can do anything" but "if you like, I'll hop into the car right now and come over", if that makes sense.
It's not so much what you say, but that you are present (be it real or virtual).