I lost my son in the same way, tragically, at the beginning of 2019. Almost all of my friends are Christians of some form or another, they all know I’m an atheist and they were all wonderful at expressing their love and sympathy without mentioning god ...not one of them, they respected that neither my son nor I believed. They may have prayed for my son and me privately, I have no doubt that some of them did, but all they did was let me know that they were there for me if I needed anything and that I could call on them for support at any time.

Last year a neighbor of mine lost her son to suicide. This is what I wrote in my sympathy card. “ Dearest Lipi, words alone cannot express my sorrow over the tragic loss of your son. I am wishing you peace to bring comfort, courage to face the days ahead, and loving memories to forever hold in your heart.

“Died by suicide” is the preferred phrasing, so it doesn’t blame the victim. But I wouldn’t refer to the matter of death in your note. I might write: “Dear Xyz —- I cannot imagine the pain you are feeling right now, but know that my heart goes out to you and your family. In time, may good memories replace sorrow. With love, Mary Jane

Aussies are a bit different. With someone I'm close to I'll say something along the lines of "this just bloody sucks. I'm here for any help that you need". And then I'd show up later to see what I could do to help, as people who are grieving deeply don't feel like doing the mundane stuff like feeding themselves.

Offer or show them that you condolences are truly heart-felt first, offer them hugs or your shoulder/s to lean on or cry on whilst you hug them, that you WILL be there for them when and where ever possible, offer to do the Grocery Shopping for them if you can
I still remember how, when 10 days after my younger sister and her husband left after we held the Memorial Service here for my 16 year old daughter that I could NOT face even going out to buy fresh bread, milk, etc, etc.
I was completely alone, lived on stale bread, toasted, powdered milk well past its use by date, about 1-2 hours of fitful sleep per night IF I was lucky, UNTIL my friends Alan and Meg Waldron knocked on my door, would NOT take no for answer, and asked me to hand over my ATM card and Pin number so THEY could go and get groceries for me.
My larders were totally bare btw.
1 hour later they returned, groceries and ATM receipts in hand, made me a meal, sat with me while I ate it, then Meg, being Meg, ORDERED me to take a long shower, I may have whiffed a little btw, change my clothes, get my "bum into gear" and "come for a ride around town" to get me out of my house for a while.
From then on Alan, Meg and their daughter, Naomi, have become far more of family to me than my own family ever have been.
Sadly, almost 4 years ago, we lost Meg to the cancers she had battling for 20+ years but she left us with a great legacy, her Grandson, my 'adoptive' nephew, Henry, and me with a whole, new, wonderful family.
Something I will never forget, be eternally grateful for and will treasure as long as there is breath in my body.
So, friend, hang in there, because what you give can so often be returned in Spades.

Though the responses below are fine if you are comfortable with them, I usually prefer to keep it as simple and plainspoken as possible: "My sympathies are with you for your loss (or in this difficult or challenging time)".

Just say:

"I'm sorry for your loss."
"You have my condolences for the death of _____."

If you've normally had regular contact with them, or even if not but want to be available, check in with them once a week. Express your willingness to listen. Bring them dinners, lunches, ready made, or even just the ingredients in a basket. Offer to help in any way they want help, whether helping with any paperwork, or helping cleaning, dishes, baby sitting. It's a tough time, and everyone responds differently. They have a big loss and it's going to be stressful.

I think you said it yourself. Just express your sorrow, but leave out the prayers and other flimflam. Speak from the heart.

Just say that your very sorry for their loss and tell them you’ll always be there for them and offer to do all you can to help them.

Sorry for the bad news. Just say 'My thoughts and heart go out to you and your family"

I usually say that I hope that their good memories help them through this difficult time. Also if they’re close I offer to help if they need any, I’ve helped someone clean their family member’s home before, which I believe was a big challenge.

What's wrong with "So sorry for your loss", and asking what you can do (being specific is helpful!)?
And maybe recounting a happy memory of the loved one in better days, if you have any. Also donations to the family and/or specified charity if requested. No religious BS is necessary on Any occasion!

Just be real. say "You are very sorry for their loss, and that you can't (or can) imagine how they are feeling, but that you are there if they ever need to talk or share their grief." Give a phone number.

I often extend to take them on a small road trip. This could be hiking, or just visiting a place like an Observatory or wildlife preserves or a natural wonder where you can ecperience something new or awe inspiring. Often people suffering from a loss need to grieve and reset from there old life to start a new one, and a reminder that their existence and experience, though small compared with Nature itself, is a part of life, but not all of life. Hopefully they see a path to a new life, this time without the ones they love.

A few sentiments come to mind:

“I’m so sorry for what you and your family are going through.”

“How can I help out?”

“Here’s some casserole / soup / chili / etc., to help you cover meals over the next few days.”

“I’m here for you if you’d like to talk.”

“How are you feeling?”

“How is your daughter holding up? Does she need anything?”

“Words can never provide the comfort that close family and friends can. I hope you can be present for one another and help one another through this difficult time.”

"I'm so sorry. I can't imagine what you (your daughter) must be going through. That's very sad."

If you knew him well talk about some things about him and what you will remember about him if appropriate.

I lost a son in the early nineties, and was completely devastated. Most of my friends and relatives were Christians and they said some of the most ignorant things "he's in a better place", "God needed another flower for his garden" and many other things similar. It was the people who just cried with me and gave me a hug that I appreciated most. Honestly though, no Matter how ignorant the words were, I appreciated all who made an attempt to be there for me. I try to do the same when someone I know loses someone.

Send your Condolences, Sorry for your loss, Wish them Well, etc.

Really? Can't you express your sorrow and compassion without a religious reference? Or is it because the sentiment is fake why you have to ask? How do I pretend to give a dam? Maybe just don't say anything like I probably should have to your question.

I'm sorry for your loss, works for me.

A prayer isn't just a religious thing exactly, itis thinking so pray. It's fine.
I'm sorry for your loss.

a heartfelt I'm sorry for your loss is always sufficient. I've donated planted trees in the person's honor (also good for environment), taken food, done chores, etc. It all depends how close you are to them.

I just give my condolences.

Simple:

My/ Our Condolences!

"Sending you our heartfelt condolences. You are in our thoughts." instead of prayers...