Close to Common-Practice
These systems require minimal changes to the usual Western fare.

Tartini-Couper
This is just like ordinary notation but with the addition of 1/2- and 3/2-sharps and flats, which usually indicate quartertones but could presumably be used for other step sizes as well. Thanks to George Secor and Dave Keenan for this image:

Maneri-Sims
This system adds accidentals to conventional notation in order to support 72-ET. Its use is advocated by the Boston Microtonal Society.

Cowell polyrhythmic notation
In his book New Musical Resources, Henry Cowell suggested using different notehead shapes to represent base-3, 5, etc. divisions of a measure along with the usual base-2 divisions we have with round noteheads. This makes tuplet brackets obsolete. Thanks to Daniel Wolf for the beautiful renders of this piece:
Fabric - New Notation [png image]
Fabric - Old Notation [pdf score]
Fabric - Cover Page [jpeg image]
CowellFabric.mid [MIDI excerpt]

Adaptive JI
This is basically the practice of assigning just intonation ratios directly to notes, along with a way to specify root changes, which function much like key signature changes for the ratios. It could in principle be used together with many of the notation systems described here. Below are two examples of its use on top of conventional notation.
Carl Lumma - Retrofit [pdf score]
Aaron Wolf - Bye Bye Blues [pdf score]

AFMM notation
Notes are simply marked up with cents deviation values from standard notation (12-ET). Johnny Reinhard suggests 1-cent precision. Deviations should generally be less than 50 cents +/-, with perhaps a few exceptions for readability.

Linear Notations
These systems draw their nominals from a single chain of identical intervals. Accidentals indicate comma inflections on these nominals.

Erv Wilson's linear notations
These support more or fewer than 7 nominals, though apparently the chained interval is always an approximate 3:2 and the number of nominals is constrained to be a MOS of the chained interval. Erv also demonstrates a clever staff to hold the notes.
On Linear Notations and the Bosanquet Keyboard [pdf paper]

HEWM
This is a family of linear notations based on a 7-nominal chain of very close (if not exact) 3:2 fifths. A set of commas is then identified, and an up/down pair of accidentals is assigned to each. HEWMs differ in the particular commas they call out for this purpose, as well as the glyphs used. Conventional notation can be seen as a HEWM with a single pair of accidentals that map to different commas depending on the musical context.
HEWM Notation [Tonalsoft enyclopedia of microtonal music theory]
von Schweinitz - Extended Helmholtz-Ellis JI Pitch Notation

Sagittal
Sagittal is a standardized set of accidental glyphs for a large collection of commas. It supports a wide variety of tempered intonations, and is said by its creators to be optimal in several respects. It is most often used in a HEWM-like manner, but permitting a wider range of fifth sizes.

Herman Miller's linear notations
These are similar to Erv Wilson's linear notations in supporting more or fewer than 7 MOS nominals, but the generator may be of any size and the period may be any aliquot part of the octave. Herman generally uses sagittal glyphs for the accidentals. See the thread beginning at tuning-math message #11629

Planar Notations
Like linear notations, but nominals are assigned to a set of pitches that can only be traversed using TWO different just intonation intervals.

Johnston notation
This system has 7 nominals, based on the standard 5-limit diatonic scale in just intonation, with accidentals for commas like 81/80. It is often criticized as being too confusing to use, but nevertheless served for magnum efforts like Toby Twining's Chrysalid Requiem, to say nothing of Johnston's own music.
Johnston Notation [Tonalsoft enyclopedia of microtonal music theory]
Fonville - Ben Johnston's Extended Just Intonation [JSTOR link]

Tripod notation
Proposed by tuning theorist Graham Breed. Designed specifically for magic temperament and generally for systems where 225:224 is tempered out. One version requires dashed lines for dummy staff positions.
MicroABC example [pdf]
Tripod Notation: A Magic Cauldron of Musical Possibilites [pdf paper]

Diatonic Notations
For any scale of 5-10 notes/octave, notes are named in order using letters A-J as necessary. These nominals fall consecutively on the lines and spaces of a staff. If a scale step pattern yields a consonant chord in different modes of the scale (e.g. 1-3-5 in the usual diatonic scale) then those chords look alike on the staff (e.g. major and minor triads). As in the above systems, comma inflections are notated with accidentals. Common-practice notation can be seen as a heptatonic diatonic notation tuned in 12-ET.
Carl Lumma's octatonic notation example [this site]
Gene Smith's ennealimmal notation
Graham Breed's decimal notation