The basic premise behind 042 is that prisoner number 042 is told he is to become a guinea pig in a daring experiment to see if death row inmates can be integrated back into every day life. A microchip is implanted that is set to detonate if 042 becomes angry (and by proxy, is about to kill someone) as a safeguard. He is then made the janitor of a co-ed high school, under constant supervision by a team of scientists led by Dr. Shiina.
What follows is a rollercoaster of events that depicts 042's struggles to fit into a society which has made it clear from the outset that he is not welcome. Along the way, he manages to touch a set of lives who will be forever changed by the encounter, and in turn he will be touched and changed by those who gather around him. In the end, the question is asked: was this experiment a success? Can a convicted serial killer be reformed and re-enter society?

While trying to bury the body of his mother deep in the woods, high school student Yuri is approached by a strange girl who offers to help him. That is... if he promises to be her "parent killing assistant." Meet Anne Freaks, a bright, attractive teenage girl normal in every way - except that she's on a mission to kill her father, the leader of the dangerous extremist cult: the Kakusei Group.

From ADV:

When popular high school student Chiko finds a cell phone, the voice on the other end tells her where she can witness a suicide! The phone doesn't stop ringing, and soon Chiko and her classmate Bando will have to race against time to save these lost souls.

Haru Takasugi deeply regrets the day he killed his adoptive father. He dies at the age of 46, only to find himself as a teenager trapped in a world between Heaven and Hell that nonetheless feels like paradise. Takasugi's fight to regain his family and friends begins! (Source: North Star Pictures, translated)

Shiki Yukitaka is desperate for money, but he makes way too little as a long term substitute teacher. While drinking one evening, a pretty woman asks if he would be willing to assassinate people for 5 million yen a kill. Although Shiki is hesitant, he grudgingly admits that he might do it, provided all the targets are serious evildoers. He soon finds that her offer is deadly serious, when she introduces him to the rest of the team and points them towards their first target. The driver, Yan, is doing this for the money as well. Shiki's partner, Kuzumi, is a cop. He doesn't appear to be interested in the money, but he seems to be driven by a deep anger.
On Shiki's first mission, things don't go quite as planned. They didn't count on their target's 18 year old daughter Itsuki being locked up in the house. She's been confined for 10 years, due to her spooky ability to read things from people's minds when she touches them. She claims to want to be their ally, and Shiki is reluctantly made responsible for her care. With the police investigating their crime, is keeping her alive too dangerous? How long will the team be able to get away with continuing their highly paid vigilante murders?
(Source: MangaHelpers)

The story is about a Detective named Akiba who crosses paths with a High School Idol named Mizuho. He was assigned to protect her. But he's also a young detective and a bit unorthodox. For the first two volumes Mizuho ends up getting involved in Akiba's other cases but eventually it progresses into a friendship (and eventually a relationship??).
[plot synopsys from Baka-Updates]

From Selene Scans:
Murakami and Nakabayashi are just policemen trying to do their jobs. When there is a bomb explosion inside Shuei Primary School, they were asked to do their jobs as policemen and check out the lead. It turns out a doctor from a mental institution called in to warn about the explosion immediately before the bomb went off. And so begins their association with the mental patient with no name who they would eventually call "Maki"
For some reason, Maki can "speak" with spirits. There are spirits all around us and Maki can communicate with them. These spirits tell Maki about deep dark things... Things that go on when no one else is looking...