Wikipedia list of Quantum Computer programming languages
(This answer is not a copy of that webpage, it's more updated and
with verified links. In some cases the author's paper or website link is
added.)
The website Quantum Computing Report has a Tools webpage listing over a dozen links, some new and some repeating the above list.
See also QuanTiki's webpage: "List of QC simulators",
for a huge list of simulators and programming languages based on:
C/C++, CaML, OCaml, F#, along with GUI based, Java, JavaScript, Julia,
Maple, Mathematica, Maxima, Matlab/Octave, .NET, Perl/PHP, Python,
Scheme/Haskell/LISP/ML and other online services providing calculators,
compilers, simulators, and toolkits, etc.
Are there certain benefits to choosing particular ones?
If you plan on using a particular quantum computer then one would
hope that the programming language developed by the manufacturer is both
best suited for that particular machine and well supported.
Choosing a language with a larger following means that there are more Forums available and hopefully more bug fixes and support.
Unfortunately that leaves some great niche products to struggle to
gain a user base. Trying to find one language that is both
powerful/expressive and supported across various platforms is the trick,
the answer is an opinion ATM.
An evaluation of four software platforms: Forest (pyQuil), QISKit,
ProjectQ, and the Quantum Developer Kit is offered by Ryan LaRose in "Overview and Comparison of Gate Level Quantum Software Platforms" (6 Jul 2018).
I will return to this answer from time to time to make updates, without excessive bumping.