$${\mathcal{L}\{f(t)\} = \int_{0}^{\infty} e^{-st} f(t) \ dt}$$
$${\mathcal{L}\{a\} = \frac{a}{s}}$$
$${\mathcal{L}\{t^{n}\} = \frac{n!}{s^{n+1}}, n > 0, n \in \mathbb{Z}}$$
$${\mathcal{L}\{t^{r}\} = \frac{\Gamma(r+1)}{s^{r+1}}, r \in \mathbb{R}}$$
$${\mathcal{L}\{e^{at}\} = \frac{1}{s-a}, s > a}$$
$${\mathcal{L}\{\sin(at)\} = \frac{a}{s^2 + a^2}}$$
$${\mathcal{L}\{\cos(at)\} = \frac{s}{s^2 + a^2}}$$
$${\mathcal{L}\{\sinh(at)\} = \frac{a}{s^2 - a^2}}$$
$${\mathcal{L}\{\cosh(at)\} = \frac{s}{s^2 - a^2}}$$
$${\mathcal{L}\{\sin(at+b)\} = \frac{s \ \sin(b) + a \ \cos(b)}{s^2 + a^2}}$$
$${\mathcal{L}\{\cos(at+b)\} = \frac{s \ \cos(b) - a \ \sin(b)}{s^2 + a^2}}$$
$${\mathcal{L}\{e^{at} f(t)\} = F(s-a)}$$
$${\mathcal{L}\{\delta(t-a)\} = e^{-as}}$$
$${\mathcal{L}\{\delta(t-a) f(t)\} = e^{-as} f(a)}$$
$${\mathcal{L}\{u(t-a)\} = \frac{e^{-as}}{s}}$$
$${\mathcal{L}\{f(t-a) \ u(t-a)\} = e^{-as} \ F(s)}$$
$${\mathcal{L}\{f(t) \ u(t-a)\} = e^{-as} \ \mathcal{L}\{f(t+a)\}}$$
$${\mathcal{L}\{t f(t)\} = - \frac{d}{ds}(F(s))}$$
$${\mathcal{L}^{-1}\{F(s)\} = f(t) = - \frac{1}{t} \mathcal{L}^{-1}\{\frac{d}{ds}(F(s))\} }$$
$${\mathcal{L}\Bigl\{\frac{f(t)}{t} \Bigr\} = \int_{s}^{\infty} F(u) \ du}$$
$${f(t) = t \ \mathcal{L}^{-1}\Bigl\{\int_{s}^{\infty} F(u) \ du \Bigr\}}$$
$${\mathcal{L}\Bigl\{\int_{0}^{t} f(v) dv \Bigr\} = \frac{F(s)}{s}}$$
Convolution:
\({\mathcal{L}\{f(t) * \ g(t)\} = F(s) \ G(s)}\)
$${f(t) * \ g(t) = \int_{0}^{t} f(t-v) \ g(v) dv}$$
$${\mathcal{L}\{f'(t)\} = sF(s) - f(0)}$$
$${\mathcal{L}\{f''(t)\} = s^2F(s) - sf(0) -f'(0)}$$
$${\mathcal{L}\{f^{(n)}(t)\} = s^nF(s) - s^{n-1} f(0) - s^{n-2} f'(0) - \dots - s f^{(n-2)}(0) - f^{(n-1)}(0)}$$