What is the utility of ADM decomposition of the space-time metric? I know it's one of the possibility of quantization of gravitational field's degree of freedom but it is introduced also in other situation. My question is what is the use for this kind of decomposition?
 A: People were excited about the ADM equations for two reasons. One was quantum gravity -- the equations give you a canonical coordinate and its momentum, so you might hope to promote these to quantum operators and be done with it. Decades later, little has come of this.
The other reason is numerical relativity -- the study of the evolution of spacetime given an initial state. The idea is to foliate spacetime with spacelike (Cauchy) hypersurfaces and specify the metric and its derivatives on one such surface. Assuming you satisfy the contraint ADM equation, you can use the evolution equation to evolve the metric forward in time. This can be done in vacuum, or with any classical field entering as source terms via the stress-energy tensor.
This idea also bore little fruit for many decades. It turns out the standard form of the equations is ill-suited for numerics, requiring reformulation such as was done by BSSN. Combine this with subtleties about horizons and trapped surfaces and how to safely excise black hole singularities and the problem becomes that much worse.
Only within the last decade or so have the fundamental problems been resolved, but now evolving spacetime numerically is done by many groups with great success. Simulations are often of two compact objects merging, where one hopes to get the dynamics correct and predict what electromagnetic and/or gravitational wave signatures would be expected.
