How can I handle divergence that appears in many physical problem? I came across with the following type of integration with singularity.
$$\int_{s_2=0}^{s_2=\infty}\int_{s_1=0}^{s_1=s_2}\left(\frac{1}{s_2-s_1}\right)^{3/2} \,ds_1\,ds_2 \, .$$ 
How can I solve it?
 A: As suggested I have expanded my comment into an answer. 
There's no general prescription for dealing with divergent integrals in physics. Typically when an integral like this shows up it means is that the integral is not the full story, but the missing pieces of the puzzle depend on exactly what it is you're trying to do.
For example in UV divergences in quantum field theory require remormalization, one says you are missing a separate infinite comtribution from the 'counter-terms', when that is added the result is finite. There are also IR divergences which are associated with massless particles, but which go away whenever you ask a truly physical question (e.g., account for the finite resolution of your detector). In electromagnetism it is possible for the electric potential to be divergent, but for the force (which is actually observable) to be finite since it's the derivative of the potential. In other contexts sometimes the argument is that your model is breaking down. A famous example is the singularity at the center of a black hole, which is thought by many to be a signal that general relativity does not apply near the black hole's center, rather than a problem that can be fixed within general relativity. 
In any case, often a good first step is to regulate the integral: write the divergent integral as a limit of a sequence of converging integrals. This can help to diagnose where the divergence comes from and how strong it is. In your exaple you could regulate by taking the $s1$ integral from 0 to $s2-\epsilon$ then sending $\epsilon\rightarrow 0$. 
