Matt/Mello fanfics often include a teen romance (perhaps unconsummated) in Wammy's House, Matt's heartbreak over Mello leaving to join the Mafia, and then later include angst over either Matt's death, Mello's death, both, or if their canon fates are avoided, their near-brushes with death. If the fanfic is non-angsty, Matt/Mello often appears as a secondary couple in a Ship Mates fanfic along with L/Light, with all four working as detectives together and getting along surprisingly well, despite Light being a (possibly reformed) serial killer.
The most common type of Death Note fic is Light and L's kinky chain adventures during the Yotsuba arc.
Light complains about L eating so much sugar and not letting him sleep. L complains about the time Light spends on personal hygiene.
Light and L forget that they are working on a high profile murder case and are paranoid neurotics and go to the movies / beach / mall / candy store / fun place for no good reason.
Light forgets that he's on a righteous crusade to eliminate evil and because of the power of true love gives up the notebook to be with L.
Mr. Yagami gets mad at L or Light for sleeping together.
Sometimes L/Light and Matsuda/Sayu are Ship Mates which leads to much banter and teasing.
L and Light team-up against an eviler, incredibly brutal OC Kira. The OC will often be 1) so brutal in his/her methods that it makes Light ill to show how much more eviler s/he is 2) a graduate of Wammy's House.
The Kira's VictoryAU where Light is so grateful to Misa/Takada/Mikami/follower of choice that sex ensues.
Rem invoked some unknown or little-used Shinigami loophole in order to spite Light so that L is still alive.
Fluff where Light hangs out with his little sister and gets a Pet the Dog moment.
There are other kinds of Notes and have the appropriate kind of god or spirit haunting them—The Life Note, The Debu Note (Fat Note), the Love Note, the Sex Note, the Yaoi Note, the Math Note, the Study Note, etc.