Note: I don't know that its possible to answer this without mentioning the things that have happened, some of which were pretty horrible. Not everyone may want to read this.
TLDR: People will be a horrible out of civilization as in it, and the rarity is matched by the severity and barbarity of the incidents.
For theft;
"We do have some crime. One of our volunteer caretakers was robbed at gunpoint a month ago at a trail shelter near Eckville, Pennsylvania," Proudman said. "The individual was captured in Wyoming."
Most of the 200-300 incidents reported a year are petty crimes, Proudman said, including vandalism, and backpacks getting stolen.
Source
Sadly, one of the biggest dangers along the trail is that posed by other people. While most hikers you come across will likely be friendly, some might be aggressive or try to steal your pack when you’re not looking. Theft is especially prevalent at small towns along the trail. Never leave your bag lying outside of a restaurant, store, or hostel along the way.
Source
Trouble is uncommon on the Appalachian Trail, but theft is not uncommon. Cars parked at trailheads may be targeted for break-ins, because thieves know that the owner will be away for a while.
Source
There is also a study done that simply asked if hikers had experiences that made them feel unsafe while hiking the trail and while not all of those involved people (such as unsafe conditions) a number of them did.
Amongst the 467 solo thru-hikers (thru-hikers who hiked less than 25% of the trail with a dedicated partner), 233 (50%) reported having an unsafe experience. There was no difference between men and women in reporting an unsafe experience.
...
Proportionately, more solo women than solo men reported their most unsafe experience as being a strange person on the trail, strange person in town, hitchhiking, and other.
...
Putting sex aside and looking at the general difference between solo hikers and partners, the odds of a solo hiker having an unsafe experience was 1.3 times more than that of a hiker in a partnership, so this implies that hiking with a partner has a slight overall protective effect, regardless of sex.
...
In fact, 50% of solo thru-hikers reported having no unsafe experiences at all.
A significant limitation of this survey was that it only asked whether or not hikers had an unsafe experience (i.e., yes or no). For those who did report an unsafe experience, the survey did not ask “how many?” So the case may be that even though women and men both reported having at least one unsafe experience at rates that did not differ, women could have still experienced more unsafe events than men, or vice-versa.
Source
For murders and rapes,
Since 1974, about ten murders took place on or near the Appalachian Trail.
Source
Nine of those are listed here.
This Washington Post article from 2008 says,
There is about one assault a year and one rape every three years, on average, according to Conservancy figures. There have been eight murders linked to the trail since the 1970s, King says.** The most recent was in January, when Meredith Emerson, 24, was abducted on Blood Mountain in Georgia and killed by Gary Michael Hilton, a 61-year-old drifter.
Source
Beyond that, there is little but news stories but not any studies that I could find.
The 19-year-old from New York paused in her Appalachian Trail hike to mail herself a package that she could pick up after she'd hiked farther north.
A Vinton man saw her walking along U.S. 11 toward the Troutville Post Office and offered her a ride in his sedan.
But he took her instead to the office of the Botetourt County trucking company where he worked, and he sexually assaulted her.
Source
Gettysburg police said a 46-year-old Alabama hiker received second- and third-degree burns in a Thursday morning assault by people who may have thought he was homeless.
...
Police said two men got out of the car, threw liquid on the victim and lit him on fire.
Source
Randall Lee Smith (1954 – May 10, 2008)5 was a convicted murderer from Pearisburg, Virginia.7 He pleaded guilty to two counts of second-degree murder in the deaths of Robert Mountford Jr. and Laura Susan Ramsay, who were killed while hiking the Appalachian Trail, in May 1981.
...
On May 6, 2008, Smith attempted to kill two fishermen less than two miles from the site of the 1981 murders.
Source
Williams, 24, of St. Cloud, Minn., and Winans, 26, of Unity, Maine, were found about a week after they set out on a camping trip in May 1996. Their bodies were discovered about a quarter-mile from Skyline Drive off the Appalachian Trail in Shenandoah National Park, south of Luray. Their hands were bound, their mouths gagged and their throats slit.
Rice, 35, has been charged with two counts of capital murder for the killings and faces two additional counts of capital murder on the grounds that he intentionally selected Williams and Winans because they were female and because they were gay. He has pleaded not guilty to the slayings. He is currently serving an 11-year sentence for assaulting a female bicyclist at Shenandoah National Park in 1997.
Source
Five of its hikers had been killed in four attacks, the earliest in May 1974, the most recent in May 1988.
Those crimes shared traits with what transpired at Thelma Marks. Two of the four attacks were aimed at couples. Three came at trailside shelters. All were ghastly.
Still, none drew the attention, or generated the angst, of the incident here. Perhaps it was because Thelma Marks fell within range of news media in New York, Philadelphia, Baltimore, and Washington, D.C., and because it involved not only a crime but a mountain manhunt that lasted a week. Perhaps the shelter’s remoteness—far greater than that of past trouble—played into our big-city uneasiness about what lurks in the woods at night.
Maybe it was the sheer savagery of the act. Or the questions that lingered when the man responsible would not say why he shot Geoff three times or why he tied Molly’s hands behind her back and looped the rope around her neck. Why he raped her. Why he stabbed her eight times in the neck, throat, and back.
Source
Meredith Emerson used her wits and martial arts training when she was attacked in the north Georgia mountains by a drifter who eventually killed and decapitated her, the convicted killer told investigators.
Gary Michael Hilton described his four days with Emerson, and how she fought him from the moment he tried to overpower her as she hiked with her dog, Ella, according to the interviews that The Atlanta Journal-Constitution obtained from the Georgia Bureau of Investigation.
Source
People are far more dangerous than any wild animal, for comparison Yosemite, which has much better recorded statistics, the wild animals have been responsible for 6 deaths with half of those from Hantavirus carried by rodents compared to 44 homicides and 70 suicides Source page 575.
In summary, theft/assault/rape/murder is rare on the Appalachian trail, at the same time incredibly horrific crimes have been committed against hikers.
This doesn't mean that one should stay home (we are all going to die regardless) or to avoid people altogether but it does mean that one does need to be vigilant, and you can't assume that anyone has your best interests in heart.
