So what actual physical event would allow us to observe the speed of gravity? I have read, if the sun disappeared it would take 8 minutes before this could be detected on Earth due to speed of gravity being $c$. Of course, the sun can't disappear. Perhaps realizing this impossibility, people write of the sun exploding instead. But if the sun literally exploded the mass would still be present for a while, just dispersed.
So what actual physical event would allow us to observe the speed of gravity? If rotating massive objects, how? And if the sun exploding would allow this observation, how so?
 A: Colliding stars and merging black holes produce gravitational waves that we have detected.  Occasionally we have been able to locate the source and correlate it with electromagnetic emission.  By comparing time of arrival of the two signals and knowing the distance to the source - and knowing the composition of the intervening space - we can compare the speed of the gravitational and electromagnetic signals.
The last section of this Wikipedia article describes such an event.
A: You need to have a changing quadrupole moment in order to see gravitational waves. A spherical explosion would not produce any gravitational waves at all, so exploding is not the answer.
What you would want is to collapse the sun into two half-suns and then have the two half-suns spin around each other very rapidly. Of course, that wouldn’t conserve angular momentum so you would probably have to make two pairs and have them counter-rotate
A: Gravitational waves propagate around light speed and have been detected by LIGO.  Unfortunately, such waves require angular frequency as well as gravity, i.e., they are a function of gravity and angular frequency.  If the angular frequency is zero, there is no strain measured, i.e., no gravitational waves, but there is still gravity.  So the question becomes, in the absence of gravitational waves, how fast is gravity?  Newton thought it's instantaneous.  Scientists currently believe that if the sun vanished, there would be a delay of around eight minutes before the earth falls out of orbit.  But you might ask, how could one make the sun disappear?  Assuming you have god-like powers, you could stretch all the sun's particle wavelengths to infinity, so the sun will have zero energy.  But notice you changed the spacetime (wavelengths) to make the sun's energy vanish.  Thus, earth wouldn't have to wait eight minutes after the sun disappears.  Einstein said matter causes spacetime to curve, but he also said curved spacetime causes matter to move (kinetic energy).  Another way to state it: Energy causes spacetime to curve and curved spacetime causes energy.  So the speed of the change in gravity is the time it takes to make the sun vanish by altering its spacetime, which I suspect can be done no faster than light.
A: The term 'speed of gravity' must refer to the time it takes the gravity at some place $B$ to change as a result of a change at some other place $A$. This is the speed of gravity waves.
The speed of gravity is built in to general relativity at a very basic level. So there is a sense in which every test of general relativity is an indirect measurement of the speed of gravity, because if the speed were different then the theory would be inaccurate at a very basic level and presumably this inaccuracy would show up in a large number of other predictions, including things like the precession of the orbit of Mercury, the bending of light on its journey round the Sun, and various timing effects. However to be sure of that one would have to propose another theory and then see in what phenomena the difference between the two theories would show up.
Of course we can deduce the speed of gravitational waves more directly by comparing arrival times of electromagnetic and gravitational disturbances, as other answers have said. The aim of the present answer is to point out that science in general does not proceed by an accumulation of separate measurements, as if we would be in doubt about everything which has not been directly measured. Rather, in science we develop a logically coherent framework called a model or a theory, and then we do whatever empirical tests we can to test the model or theory. It is a little like visiting the doctor: to find out about your nervous system the doctor won't need to examine every nerve, he could just hit your knee with a hammer and the response would already reveal some useful information. It is the same with things like determining the speed of gravity: long before gravity waves were detected the speed of such waves was already inferred with high confidence from the large number of precise tests of general relativity, together with the fact that that theory has a high degree of elegance and mathematical beauty, so that it cannot easily be modified to produce other theories which would pass the same tests while having some other speed of gravity waves.
