Gardner Read’s Source Book of Proposed Music Notation Reforms (on Amazon) lists hundreds of proposed reforms dating from 1702 up to 1987.  Most proposals focus on the issues of inconsistent staff lines and key signatures.  Fewer proposals addressed the issues with rhythm definition.  What is clear from this collection though is that the difficulty of reading sheet music has been recognized for a very long time.  Yet none have succeeded in attracting a really wide audience in Western Music.  At least one of the barriers that must be overcome is the need to reproduce the vast inventory of Traditional Notation sheet music in a new notation.   As daunting as that may be, the larger barrier is just bucking tradition.

The Music Notation Modernization Association - MNMA 1985-2007

To quote the Overview page of the MNMA website:

the MNMA “was founded in 1985 by Thomas Reed as an international nonprofit organization dedicated to modernizing/improving music notation. The goal was to address the shortcomings of traditional music notation that ‘make reading music substantially more difficult than necessary, pose unwarranted hardships for musicians, and prevent many young people from becoming fluent music readers.’  The membership of the MNMA included musicians, teachers, composers, and notation inventors from around the world.”  

The association mainly focused on the first two issues with Traditional Notation: inconsistent staff lines and the use of key signatures.   And for the most part, all of the designs are chromatic (all twelve notes in the scale have an equal position on the staff).  Hundreds of Alternative Notations past and present have been examined, catalogued, and presented on their website for public review. 

The MNMA conducted a research project to evaluate what it judged to be the leading Alternative Notation contenders by a small group of highly experienced musicians.  The evaluation criteria were heavily weighted towards advanced player needs.  In spite of the more logical organization of the candidate designs, this group of advanced players had no unanimous positive support for any of them.  Most found that Traditional Notation was a better solution.  In my opinion, this is most likely because they had been reading it over their entire musical careers and did not see any advantage in using another notation.  

Thus, in my own interpretation of the results, longtime readers of Traditional Notation have no incentive to use, much less switch to, alternative systems even though they are more logical and consistent.  And why should they if they can already read Traditional Notation?  (It's an open question, though, whether other musicians might adopt an Alternative Notation as a "2nd language" for certain features such as visual identification of intervals or graphic note duration.) 

Unfortunately, the MNMA research project did not include evaluations from the perspective of a beginning student of music.  Given a choice, I think new students who are unbiased by a history of reading Traditional Notation are more likely to choose a different approach if it is easier to read.  Bottom line: I think it is more likely that Alternative Notations will be adopted by new students of music rather than experienced readers of Traditional Notation.  So throughout the discussion here, new students are assumed to be the target audience (with a few small exceptions).

The MNMA has been inactive since 2007, but the follow-on Music Notation Project forum is available on the same MNMA website to continue the discussion of Alternative Notations.

Others

Since Gardner Read’s book was published there have been a number of US Patents filed for Alternative Notations. I am aware that there is a researcher who is investigating these.  I am not aware of any that have achieved any significant success in being adopted on a wide basis.