Now that the dust has settled on the 2011 superluminal neutrino debacle at OPERA, I'm interested in understanding the current status of experimental searches for neutrinos. Although the OPERA claim was a fun puzzle for theorists trying to find explanations, the claimed scenario wasn't anything like our current best guess as to what to look for in a search for tachyons. E.g., we do expect tachyons to be electrically neutral (pace Baldo 1970), but we also really expect them to have spin 0.
The only particle-physics test I've been able to locate information about is Alvager 1968. They searched for production by photons on lead and put an upper limit on the cross-section. The experiment would only have been sensitive to charged tachyons. This is extremely old, predates the modern understanding of how tachyons fit into QFT, and is of less interest than searches for neutral tachyons, which are what we actually expect to exist.
There is a big bibliography here, but it mostly focuses on the OPERA debacle.
Recami 2009 is a recent review article, but I don't have access to the conference proceedings, and the article doesn't seem to be available online.
Tachyons are expected to exist as waves that are either localized or superluminal, but not both (Shay 1977, Baez). So it seems like superluminal propagation of a wave packet, as in the OPERA result, is not really the right experimental signature to look for. What is the right signature? Non-electromagnetic Cerenkov radiation?
If they're expected to be nonlocalized, does it make more sense to look for cosmological evidence, as in Davies 2012?
It seems tough to define an appropriate way to search for tachyons, since QFT basically says they shouldn't be able to exist in the naive classical sense envisioned 50 years ago. On the other hand, the willingness of theorists to try to explain the OPERA results suggests that there is wiggle room. If you look at a freshman text like Tipler 2003, he's still referencing Alvager 1968 as the null experimental result on tachyons. This seems unsatisfactory. I can't believe that there haven't been better direct searches in the last 45 years.
This question is similar to, but much more specific than, a previous question: Tachyon and Photons
Alvager and Kreisler, "Quest for Faster-Than-Light Particles," Phys. Rev. 171 (1968) 1357, http://adsabs.harvard.edu/abs/1968PhRv..171.1357A
M.Baldo, G.Fonte & E.Recami: “About charged tachyons”, Lett. Nuovo Cim. (first series) 4 (1970) 341-247. http://dinamico2.unibg.it/recami/erasmo%20docs
P. C. W. Davies, Ian G. Moss, "Cosmological bounds on tachyonic neutrinos" http://arxiv.org/abs/1201.3284
Recami, "Superluminal waves and objects: An overview of the relevant experiments," Journal of Physics: Conference Series, Volume 196, Issue 1, article id. 012020, 14 pp. (2009)
Shay and Miller, "Causal and noncausal propagation of both tardyon and tachyon wave functions," Nuovo Cimento A 38 (1977) 490, http://link.springer.com/article/10.1007%2FBF02730018
Tipler, Modern Physics, 2003
[EDIT] I did succeed in finding some references more recent than 1968. I don't know if these represent the state of the art, but to avoid wasting other people's time, I'll list them here:
Clay, A search for tachyons in cosmic ray showers, http://adsabs.harvard.edu/full/1988AuJPh..41...93C Australian Journal of Physics (ISSN 0004-9506), vol. 41, no. 1, 1988, p. 93-99. -- cosmic rays, time of flight
Baltay, C., G. Feinberg, N. Yeh, and R. Linsker, 1970: Search for uncharged faster-than-light particles. Phys. Rev. D , 1, 759-770, doi:10.1103/PhysRevD.1.759. -- accelerator experiment
The following are not experimental papers, but do give helpful surveys of methods of searching for tachyons:
Recami, "Superluminal Waves and Objects: Theory and Experiments. A Panoramic Introduction," ebooks.iospress.nl/Download/Pdf/28897
Sudarshan, "The nature of faster-than-light particles and their interactions" http://wildcard.ph.utexas.edu/~sudarshan/pub/1969_005.pdf