Does the dirac delta function have a Laplace transform?

[AdrianZ]

If yes, how can we find it?

 

[LCKurtz]

If yes, how can we find it?

The δ function isn't actually a function; it is a distribution. It can be thought of informally as the limiting case of a pulse$$
f(t) = \left \{\begin{array}{rl}
h, & 0 < t < \frac 1 h\\
0,& \frac 1 h < t
\end{array}\right.$$ Then, again informally, you can calculate its LaPlace transform by working$$
\lim_{h\rightarrow \infty}\int_0^{\frac 1 h} he^{-st}\, dt$$ You should get an answer of 1.