"Happy Birthday," said Jill.
Maureen's face clouded over.
"I didn't want anyone to know."

OR

"Happy Birthday," said Jill. Maureen's face clouded over.
"I didn't want anyone to know,"

It's the first one isn't it? Even if you've done a big bit of narrative before it you've got to put the bit about who is speaking just before it, right? I'm sure that's right - just had a brain freeze. Sorry and thank you. (Could always look in a book, I suppose...)

Ooo-er. I think it's:

"Happy Birthday," said Jill.
Maureen's face clouded over. "I didn't want anyone to know."

But I could be wrong.
Susiex

I'd tend toward Susieangela's version. The key point being to separate the actions that apply to each of the characters for the benefit of the reader's understanding rather than any hard rules.
G

Oh lordy - it probably is. Thank you very much Susie, will now go back and correct everything!
What happens then if:

Jill smiled. "You're just being silly."

That would then go on a new line, wouldn't it? Sorry and thank you.

Yes, new line as you've changed character focus. (Again, for ease of comprehension more than anything else.)
G

Like Gaius, I don't think there are rules, really. It's just how the sentences make sense visually to the reader.

"Happy Birthday," said Jill.
Maureen's face clouded over. "I didn't want anyone to know."
Jill smiled. "You're just being silly."

The only slight thing is that you're then beginning to repeat the phrasing - ie: first you describe the facial expression, then what they say. It might feel better to rearrange slightly:

"Happy Birthday," said Jill.
Maureen's face clouded over. "I didn't want anyone to know."
'You silly. Why ever not?'

Just playing around. You don't really need to tag Jill in her second sentence because you've already established it's a conversation between them.

Susiex

Yes, new line when a new character speaks, but if that new character's action comes before their speech, new line with the action.

"Happy Birthday," said Jill. Maureen's face clouded over. "I didn't want anyone to know." Jill smiled. "You're just being silly." Maureen got up as if she wanted to leave. Jill took a deep breath. "Why aren't you proud of your birthday?"

Horrid extension of yours, FV, purely make the point - sorry. I see a lot of writing where it breaks after 'silly', and I then find I assume that 'why aren't you proud' is spoken by the other character, when in fact it's still Jill...

Emma

Thank you so very much, everyone - that's exactly what I needed to know. New character, new action.

I posted some work and to my horror my speech tags had gone all over the place and no one could understand who was speaking - as you suggested Emma. So thank you for this, very much. Hope your writing is all going well today.