Talking About Sensitive Topics - Need more examples Please

Hi UncleA - I'm sorry we confused you with the two sets of instructions.  
 
There's a difference between a casual Monday morning: "Hey, have a good weekend, do anything interesting?"​and "Did you do something with your husband over the weekend?​"
 
One's a casual open-ended question, and the other assumes that someone a) has a husband and b) wants to do something with the​m.  ​
​If they say​, "I went line-dancing", then "oh, that sounds like fun!" is a continuation of the conversation, without asking prying questions about who they were were with.

 
The other part I think you're missing (which means we didn't make it clear enough, not that you missed it) is paying attention to what people say to you.   For instance, if none of your directs have ever had to take a day off because a child was sick, then perhaps they don't have school age children.  
 
If they do take the day off, when they come back, you can say "I do hope your child is feeling better".  Your direct may or may not give you more details, and that's their choice.  They'll probably at least say "yes, he's feeling more himself", at which point you know they have a boy!
 
If they have a photo on their desk, you can say "what a lovely picture".  (Don't assume the relationship between the person and the people in the picture).  Most people will tell you who they are.  You can do this now, even if the picture's been there for a year.  Just say, "Every time I come over to your desk, I think, what a lovely picture and I realize I've never complimented you on it". 
 
People often tell you where they grew up, what they like to do, who lives with them in oblique ways.  Pay attention.  Most people don't hear them, because they're too busy thinking about themselves.
 
Finally, there is one really good opportunity to find out about people - when someone new joins your team.  Go out to lunch and have your current team tell the new person something about themselves.  You can start, and if you include your family and hobbies, others will follow suit with as much as they feel comfortable with.  You can do this if your team is working with someone new in a different team, or a contractor or temp starts with you - it doesn't need to be something permanent.   It's better if this is in an informal setting, but if you have to do it on a conference call, you can do that too.
 
Does that help?
 
Wendii