There is no grief like the grief that does not speak. - Henry Wadsworth Longfellow

A coworker is returning to work after the death of a loved one. You want to offer your help and support, but you’re worried about offending or upsetting him. What should you avoid saying? What should you avoid doing?

Don't stay silent. Saying something is better than saying nothing. If you have no idea what to say, say that. Tell your colleague you don't have the right words to convey how sorry you are, but you are willing to help, support and listen whenever she needs you. If you don't acknowledge her loss in some way, she may assume you don't care, not that you can't find the right words.
Don't assume he'll ask for help. He may be embarrassed to tell you that he can't get through a client call without crying or that sitting alone in his office analyzing the company's financials makes him anxious. Vague offers (e.g., Let me know how I can help ...) are worthless. He may not know what help he needs or have the energy to ask. Take the initiative, and be specific about how you can and are willing to do.
Don't try to fix anything. You don't have that kind of power. There's nothing you can say or do that will bring your coworker's loved one back or change the circumstances of the death. Listening and witnessing her grief are what will help the most.
Don't ask about his loved one in front of other people. Mentioning his loss in the weekly team meeting could prevent him from concentrating and undermine his efforts to prove to his boss that he's ready to be back at work. Save the kind words for when it's just the two of you, but first make sure he's not heading to an important appointment.
Don't take anything personally. If you ask how she's doing today, and she says 'Fine' in a clipped voice, that doesn't mean she's brushing you off. She may be afraid to start thinking about everything that's happened for fear she'll break down at work. If you offer to go for a walk or to take her for coffee, and she says 'No,' her refusal may have nothing to do with you. She'll appreciate your kindness even if she doesn't take you up on your offer.
Don't assume everything non-work related is covered. Not everyone has a solid support system. His family may need meals. He may lack the energy to mow the lawn or find it too painful to go to the grocery store. Ask.
Don't talk about your losses. The death of your grandmother isn't comparable to the death of your colleague's spouse. Nor is the death of your dog. Even with the best of intentions, it's easy to find ourselves playing the "Grief Olympics" to determine whose loss is more painful or become the subject of a conversation that should be about your colleague's circumstance.
Don't assume she's "fine," even if she says she is. Maybe she's just trying to get through the day, finish a project or hold it together for a meeting. Even if the intensity of her grief is easing, and she's finding it easier to be at work, that doesn't mean she's back to normal or that everything is going well. Keep checking in and offering to help.
Don't assume his closest friend in the office is providing all the support he needs. Sometimes even our closest friends don't lean in. Find out for yourself if he needs additional help or someone else to talk to.
Don't assume she'll get over it in a few months or even a year. People grieve for a lifetime. You may not need to keep offering help or ask how she's doing as frequently. But keep checking in and say her loved one's name whenever appropriate.

Don't Say This. Don't Do That.

MARGO FOWKES

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