How precise are the observational measurements for the speed of gravity? General Relativity says that gravity moves at the speed of c.  How precise are the measurements?
The accuracy is extremely good thanks to LIGO and a binary neutron star merger.  It's accurate to about 3 parts in 1 quadrillion!
See the paper Gravitational Waves and Gamma-rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A
The paper constrains &quot;the difference between the speed of gravity and the speed of light to be between $−3×10^{-15}$ and $+7×10^{−16}$ times the speed of light.&quot;
Older response before the LIGO observation:
One of the more recent of gravity was first measured in 2003 using Jupiter and a quasar.  The measurements were for the purpose of proving a finite speed and weren't very precise.  See the paper for more.
More recently in 2013 a Chinese group built a model using Earth's tides that helped them narrow it down.

... [T]he speeds of gravity are from 0.93 to 1.05 times the speed of light with a relative error of about 5%. This provides first set of strong evidences to show that the speed of gravity is the same as the speed of light.

This is so far the most accurate measurement I've seen.  See the paper for more.
In the near future, LIGO may be able to provide more accurate measurements by comparing the distance among detectors and the delay of observation.
Also related, see the question &quot;How is the most accurate value of  G  measured?&quot;