First, cats do not "run away" or "stray." They investigate new places and find themselves trapped, they get lost if driven away from their territory, they are spirited off by circumstances beyond their control, they become ill or injured and creep away to a quiet dark place. But they rarely voluntarily leave their home, even if badly treated. It is imperative when a cat turns up missing to begin an aggressive search immediately. Your cat's life could depend on it.
Try to think like a cat, your cat specifically. Look around and try to imagine what could have happened to account for the disappearance and help you look in the right places.
Cats are excellent hiders. Look first in your immediate area. Check the house carefully. Then check again, even in those spaces where you are certain a cat could not fit (chances are, they can). Listen for sounds of distress (cats explore weird spaces and can be fatally attracted dark places and to vertical spaces -- look behind the water heater, for example, or in that closet or cupboard or attic you accessed a while back). If there have been changes in the environment lately, with nifty new places for a cat to explore, look there. Don't forget to look in your own backyard in case the cat has been injured or gotten stuck somewhere.
Then go door-to-door, taking a pad and pen with you to write down your name and number. Better still, first make up a quick poster with a brief description, a clear photo, and the cat's name and your name/phone and run copies at the nearest photocopy center -- be sure to make your phone number (or at least the "lost cat" part) prominent enough to be seen by a passing car.
Ask your neighbors to look and listen for a cat in their area. Ask them to check their garage or other outbuildings, look in their trees, check their basement. And ask them to call your cat's name and listen carefully for signs of distress. Leave your poster or name/phone with and a description and name of the cat. Tell them you may check back with them later. Then ask permission to enter their property to look for yourself (the cat might be too frightened to respond to a stranger). Most people will gladly cooperate.
Ask neighbors if they have noticed a "new" cat in the area, even if they think it belongs to someone else. Sometimes people "find" cats or kittens and decide to keep them, either assuming they are "stray" or that they are not likely to be claimed by an owner. Children sometimes "find" new pets in this way and carry them home, where the cat is either taken in or put outside by the parent to find its way home again.
Don't rule out neighbor malice. Neighbors, even landlords, sometimes snatch cats and dump them in another neighborhood or worse. It is worth visiting shelters out of your area. It is also worth asking neighbors if they know of anyone in the area who might be trapping cats or who has a history or the potential of wishing cats harm. Be diplomatic.
Check the streets and alleys. An injured animal may not be able to get home or may choose to withdraw into a quiet place. The sooner the cat can be gotten to emergency care the better its chances of survival.
Sometimes cats climb into moving vans or parked cars and are not found immediately. Was such a vehicle in your area at the time of disappearance?
Ask neighborhood kids if they have seen anything. Give them your name and number. Kids can be a great source of neighborhood goings-on.
Talk to your mail carrier and give him/her a flyer or a photo with the cat's name and your name/phone on the back.
Now put up posters around the neighborhood and take them back to those neighbors you have already contacted if you weren't able to supply a poster earlier. Leave posters at vet clinics, local shelters (even those out of your area), and all over the neighborhood. When putting up your posters don't forget to check the "found cat" posters. At the vet's ask if an injured cat was brought in as a "stray" and ask for a description. Vets will stabilize injured cats before they are taken into the animal shelter.
Place a "lost cat" ad in both the city and the neighborhood papers. Also check the "found" ads daily.
If you offer a reward, beware the hostage-taker or bogus calls. (I met a couple whose bulldog pup was held hostage for $100 when they had offered $50 reward; they paid out of fear for the pup's life.)
Check Animal Control frequently (every other day or at least every third day) and be prepared to go down and look at the animals in the kennels; I have heard cases of the lost animal being in the shelter but not reported to the owner (by error, not design). Your description may not be sufficient to help an attendant identify the cat over the phone. Leave a photo at the front desk and ask to visit the quarantine area for sick and injured animals (sometimes overflow animals are caged there as well). Ask if any overflow cats are being held in the dog area. Also, sadly, review the DOA list. If the cat had collar i.d., its chances of being returned to you if found are much great, but don't count on it; the collar could have been lost or even removed. Microchipping is much more reliable. Bear in mind that some citizens are loathe to take a found animal to the shelter right away and will keep it for several days or longer before turning it in.
Consider using a trained tracking dog. Contact local obedience class teachers and inquire about hiring someone with a dog with Utility Dog certification to locate the cat by scent.
Don't give up. Keep looking in those same old spots, calling and listening. Try new spots; enlarge your search-area to the next block or the next after that. Don't be embarrassed and try not to let yourself become paralyzed with grief and anxiety (I know the feeling). Cats are tough customers and can last many days without food or water. They also can hide very well, remember, and may not be rescued by animal control or a citizen for many weeks after their initial disappearance.
Last word of advice is DON'T GIVE UP THE SEARCH TOO SOON. Don't give up as soon as the cat fails to return home or after only a few days and don't just wait for the cat to come back or not. Keep looking and keep checking.
One thing to add: if you've moved recently (within the year) go back and check your old place. My cat went back to our old apartment twice in 4 months, and then had trouble finding his way back to the new place, so he just stayed. He was on the roof of the old apartment, begging to be let in the attic window (that is how he used to come in and out of the apartment when we lived there) and the new tenants thought he had rabies because he was so insistent. The second time he ended up there no one was home and we had to climb up a tree (three stories) to rescue him--then scurry through the apartment building hoping we wouldn't get caught by neighbors coming home.