Looking for evidence
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While searching for your lost pet, you will always need to be looking for evidence. This means paw prints that could be your pet's, or if you notice other tracks in the area that do not match that of your pet, whether they are a cat or a dog, you will need to be attentive to other animal tracks, and who they might belong to. This can help aid you to know what is in your surrounding area, and may be an indication as to why your pet may have just disappeared or vanished with no sightings. 
Here are some examples of paw prints that you need to be familiar with.

                                                       
                     Cougar  paw print                                            Dog print    vs.   Coyote                    

In conjunction with looking for animal tracks, another clue to what wildlife is in your area, is to look for animal scat. Here are some common wildlife scat pictures.

                 
                              Coyote scat                                        Fox scat

A little more on coyotes attacking pets
Coyote predation can usually be distinguished from dog or coydog predation by the fact that coyotes partially consume their victims. Tracks are also an important factor in distinguishing coyote from dog predation. Coyote tracks tend to be more oval-shaped and compact than those of domestic dogs, plus, claw marks are less prominent and the tracks tend to follow a straight line more closely than those of dogs. With the exception of sight hounds, most dogs of similar weight to coyotes have a slightly shorter stride. Coyote kills can be distinguished from wolf kills by the fact that there is less damage to the underlying tissues. Also, coyote scats tend to be smaller than wolf scats.
Although it is rare for coyotes to attack humans, coyotes are often attracted to dog food and animals that are small enough to appear as prey. Items like garbage, pet food and sometimes even feeding stations for birds and squirrels will attract coyotes into backyards. Approximately 3 to 5 pets attacked by coyotes are brought into the Animal Urgent Care hospital of South Orange County each week, the majority of which are dogs, since  typically do not survive the attacks. Scat analysis collected near Claremont, California revealed that coyotes relied heavily on pets as a food source in winter and spring. At one location in Southern California, coyotes began relying on a colony of feral cats as a food source. Over time, the coyotes killed most of the cats and then continued to eat the cat food placed daily at the colony site by citizens who were maintaining the cat colony. Coyotes will usually attack smaller or similar sized dogs, though they have been known to occasionally attack large, powerful breeds such as Rhodesian Ridgebacks and Rottweilers. Even with a size advantage, large dogs are usually at a disadvantage against coyotes in physical confrontations, due to the fact that coyotes have larger canine teeth and are generally more practiced in hostile encounters.
 

There are many things you can be looking for as you are searching for your pet. One being fur caught on fencing or on the face of a hole leading under a building or house.

          
 
Clumps of fur that are found on the ground could indicate cat fights, or confrontations with other predators.

       

If you find evidence such as bones, fur, or excrement that may match your pet, there are some other sources that may be able to help you with this.

Forensic DNA Testing for pets. This is so cool! Check it out. http://www.vgl.ucdavis.edu/forensics/index.html

Forensic Anthropologist work with skeletal remains. http://en.wikipedia.org/wiki/Forensic_anthropology

If you live near roads, it is also advisable to checks sides of the road, especially if you see black skid marks on the road.

Use your senses, smell for decomposition or decaying odors. Look for evidence, and the gathering of flies and buzzards. Listen for cries, meows, barking cat fights, and predator calls.

Learn more about preserving evidence samples >> here

Action step - Sightings of your pet

References:
Dog vs. Coyote image http://www.conservation.state.mo.us/manag/coyotes/tracks.html
Coyote scat image © Photo by Tim Vechter, http://www.westol.com/~towhee/redtail.htm
Fox scat & Cougar print courtesy of Wikpedia.
Coyote information courtesy of Wikpedia.