In a mystery we need to see the crime within the first few pages, certainly within the first three chapters. You want to grab your readers right off the bat so start with a bang, kill somebody, blow up a bridge, or have two jetliner nearly miss one another. You need to hook your reader immediately or they may not bother to read the book. Many readers especially of mysteries choice a books by the cover, the back blurb and the first page. So grab him by the throat and don't let go.
Mystery readers like action and they like it fast. The worst thing you can do is start your story slowly and build. You don't have time! Noah Lukeman in his excellent book, "The First Five Pages" states that and editor will buy your book based on the first five pages of your novel, possibly the first page and probably the first paragraph! Scary I know, but that's the publishing business.
Don't think it's possible? Sure it is! I can't remember the title of the book nor its author, but here paraphrased is one of my all time favorite openings.
"Besides the dead man, there were four of us in the room."
Now, are you going to say, ho-hum, I guess I'll put the book down and get a cup of coffee? I don't think so! You'll want to know: who's the dead man, how did he die, who are the other four men and what are they doing in that room? Isn't that a better beginning then this?
"Sally Johnson lived all her life in Detroit. She went to school with the same group of girls and grew up with four brothers. By the time she was ready for high school, she knew exactly what she want from....closed eyes... sleep....loud snoring.
We might need that information later, but first we have to have the crime. You can fill in the back story later - but in small chunks, please.
Here's another example from my own book, "The Worst Evil."
"Chicago, February 1964
It was the bitterest winter in a century. Frigid air swept down from Canada. Hard icy winds blew off the lake. The temperatures hadn't been above freezing for weeks. Most people were cold, many were miserable. Some died."
Aren't you just a little curious as to who died? And how? And....why?
It may take you a while to get just the right beginning. It might take you months, but don't worry about it. Get it down the best you can because you know you're going to have to re-write and you can always come back to that part. Starting is hard, but don't get stuck there. I've known too many writers who agonize for months, even years on getting that first chapter perfect. Don't do that. It will never be perfect, just do the best you can, and move on. Finish writing a first draft, first.
You now have a super grabber beginning and that's great. Be sure to also introduce your detective and your villain. If the bad guy enters too late in the book, your readers will feel cheated because they didn't have sufficient information to solve the crime. And for heavens sake, don't have the villain enter in the last few chapters. People read mysteries to try and figure out who the culprit is, so he has to be there pretty much from the beginning.
The crime must be plausible. Your reader will feel cheated if the crime is something that couldn't really happen.
Have your detective discover clues and let us know what they are so we can "detect" right along with him. They have to use conventional scientific methods and do the research to make sure you're giving the reader correct information. Not sure how to find the correct procedures? Call the local police, sheriff, FBI or whomever you need to. Believe me, they're used to hearing from writers and they want us to get the facts straight. I really hate to read mysteries that have too many coincidences. Let your detective find the clues so we can see them too.
The villain should be capable of doing the crime and have sufficient motivation. It would be hard to believe that a 70-year-old, slight, preacher's wife was an axe murderer. Or that a nine-year-old boy was a serial killer. Both are certainly possible, but highly unlikely and I doubt your readers with buy either.
Your reader is trying to solve the crime along with the detective so don't have improbable situations or supernatural solutions. Of course you want to hold off as long as possible to reveal the bad guy. Make it hard for the detective to solve the crime and it will be hard for your reader to solve the crime too. He'll love you for it - and buy your next book.