Bereavement advice please

1. Yes, mention it. Elsewise, you're sending the message that the loss doesn't matter and it's not ok to mention it. The loss totally matters and will dominate that person's life for a good while, in all likelihood.

2. "I'm so sorry to hear about...your mother, Janet, whomever", said with a get-away option in the event the person brushes you off.

My favorite book about grieving is When Children Grieve, by John James. He has a similar book for adults but I think the children's book is more complete.

I think that the least you can do is send a card/email acknowledging their loss and your sympathy for it. Just the fact that you thought to send a note is sometimes enough.

For closer friends and such I like to go to the wake, if there is one. Even if I didn't know the deceased, I'm there to support my friend.

I disagree entirely with not mentioning it at all.
Obviously, everyone is different and some people don't want to talk about it at all, but I suspect it is entirely, always, appropriate to say something like: "Friend, I just wanted to tell you that I am truely sorry for your loss; I have (or have not) been there but I know it just plain sucks right now. If you want to talk or if there's anything I can do, I will.

I think the best way to handle it is to tell them that you are sorry for their loss, and let them know you are there for them if they need you. Maybe ask them if they want to do something outside of work, like get some coffee or go out to eat. You can talk to them there, and it doesn't have to be about the death. Most of the time people are relived to have a little bit of a distraction and keep their mind off of it for a little bit. Make a closer friend in the process. Don't be pushy, but just let them know you're there.