It is not possible, even in principle, to unambiguously detect a change in the speed of light over time, and this was true even before the SI units were redefined so as to give $c$ a defined value. The trouble here is that $c$ has units, and if a fundamental unitful number seems to change over time, there is always an ambiguity in whether the number is changing or the standards used to define the unit are changing. It is only possible to unambiguously detect a change over time in a unitless constant. For example, it's quite conceivable that the unitless fine structure constant $\alpha=e^2/\hbar c$ has changed over time, and there have even been claims that this has been detected.[Webb 2001] This claim is probably wrong, since later attempts to reproduce the observations failed.[Chand 2004] Rosenband et al.[Rosenband 2008] have done laboratory measurements that rule out a linear decrease of alpha with time large enough to be consistent with Webb's results. Even if Webb's claims are correct, this is not evidence that $c$ is changing, as is sometimes stated in the popular press. If an experiment is to test whether a fundamental constant is really constant, the constant must be unitless.[Duff 2002] If the fine-structure constant does vary, there is no empirical way to assign blame to $c$ as opposed to $\hbar$ or $e$. John Baez has a nice web page discussing the unitless constants of nature.
J.K. Webb et al., 2000, "Further Evidence for Cosmological Evolution of the Fine Structure Constant," http://arxiv.org/abs/astro-ph/0012539v3
J.K. Webb et al., 2010, "Evidence for spatial variation of the fine structure constant," http://arxiv.org/abs/1008.3907 ; Phys. Rev. Lett. 107, 191101 (2011)
H. Chand et al., 2004, Astron. Astrophys. 417: 853, http://arxiv.org/abs/astro-ph/0401094 ; See also http://arxiv.org/abs/0711.1742 , http://arxiv.org/abs/0905.1516
Srianand et al., 2004, Phys.Rev.Lett.92:121302, http://arxiv.org/abs/astro-ph/0402177
Duff, 2002, "Comment on time-variation of fundamental constants," http://arxiv.org/abs/hep-th/0208093
Rosenband et al., 2008, 319 (5871): 1808-1812, http://www.sciencemag.org/content/319/5871/1808.abstract