How to detect a wormhole? Question
A wormhole is a possible solution to Einstein field equations. As far as I understand, the fact that they are a possible solution does not mean they actually exist.
Are we looking for wormholes? By what methodology can a wormhole be detected?
My level of understanding
I am a geneticist and I only have rudimentary understanding of General relativity.
I understand that gravity is not a force and that an object that orbits another one actually goes a straight path in the space-time continuum. I understand that spinning, non-spherical objects (such as two black holes) create ripples in the space time continuum. I understand that we can detect such ripples by "splitting a ray of light in two" and let these rays travel back and forth in two different tunnels and watch whether their waves combine additively or destructively. Variation over time in how these waves combine mean that spatial dimensions are getting stretch and squeezed relative to each other.
 A: While wormholes might create gravitational waves if they wobble, this is likely rare and since most wormhole geometries are very unstable likely to lead to a non-wormhole situation soon. But just like black holes wormhole spacetimes bend light rays (but in a different pattern). This suggests that gravitational lensing is the way to see a remote wormhile. 
There have been papers analysing gravitational lensing from wormholes (example, example,example, review, original classic paper). If you monitor a distant galaxy a randomly passing object in front of it may pass across a star in such a way that its luminosity increases and decreases in a characteristic way. While most such efforts have focused on finding gravitational lensing due to massive objects like black holes, they have presumably also kept a small eye open for other weird patterns. There have also been proposals to look through the data for particular wormhole models (one problem with wormholes is that there are many different hypothetical types, each with their own particular lensing). 
A: This is an interesting question, and I would like to see other people's ideas on this. There may be discussion of this in the literature. The following are just my thoughts, working from first principles.
If highly non-Newtonian objects like black holes, naked singularities, and wormholes form through natural processes in our universe, then we expect them to form in runaway gravitational collapse. Once they form, they are extremely compact, and therefore we can't expect to resolve them in telescope images, except perhaps in the most favorable cases such as Sag A*. With wormholes, there is also the problem that they are expected to be unstable. Therefore, I would think that the chances of detecting a wormhole would be best if you try to detect its formation, not to detect it long after its formation.
For several decades, the standard expectation has been that the collapse of massive stars leads to the formation of a black hole. There have also been suggestions that it could lead to the formation of a naked singularity, and there have been serious proposals to observe such a thing (Joshi et al., "Distinguishing black holes from naked singularities through their accretion disk properties," apr 2013, https://arxiv.org/abs/1304.7331 ). One big difference between a black hole and a naked singularity is that a black hole is expected to radiate away only a small percentage of its mass-energy before the rest is trapped behind the event horizon, while in the case of a naked singularity it could be 100% radiated. There is also a high-frequency cut-off in the radiation spectrum when a black hole is formed, but not a naked singularity. I would think the formation of a wormhole would be similar observationally to the formation of a naked singularity in these respects.
There are theorems in general relativity that place tight constraints on the formation of wormholes by gravitational collapse. It requires a change in the topology of space, and theorems by Geroch and Borde say that you can't have topology change unless there are both exotic matter and violations of causality. (In technical terms, there has to be a violation of the weak energy condition, and the spacetime has to violate causal compactness.) Although the weak energy condition is violated by dark energy, we have no evidence that it is violated by any of the forms of matter -- typically baryonic matter and maybe dark matter -- that would be likely to contribute to astrophysical collapse. For this reason, if we are to have any chance of observing the formation of a wormhole by gravitational collapse, then we are necessarily talking about observing a process that falsifies classical GR in a regime where we had previously believed that it worked just fine. In this respect, it is very different from the observation of a naked singularity, which would be far out but not incompatible with GR.
A: The concept of Einstein-Rosen (ER) bridges or wormholes are not fully understood by physicists because it is about two blackholes (we are already having troubles with one). There are still debates going on because of the lack of observations. Especially in an information basis, it is really troublesome because these solutions of Einstein Field Equations are something about the General Relativity while information is a concept of quantum mechanics. And these two are incompatible theories as you might have known.
Now, any blackhole with a given surface area is strictly identical to any other with the same surface. So, this makes blackholes distinguishable only by their areas or radii (if it is stationary). So, any two blackholes that are connected (entangled) by a ER bridge would not give any information about it.
However, there is a big hope that ER bridge is equal to the concept of entanglement (Einstein-Rosen-Poldowsky conjecture). If it is so, then we can observe some other properties for a couple of blackholes and see if they are entangled or not. For example probably the angular momentum of two entangled blackholes would be highly correlated. 
