Fourier transform of exponent: Delta pulse or hyperbola?

Why do some tables say that Laplace (or Fourier?) inverse of exponential is a time-shifted delta pulse

$$\begin{align} \delta (t) &\overset{\mathcal F}{\Longleftrightarrow} 1\\ \delta (t-t_0) &\overset{\mathcal F}{\Longleftrightarrow} e^{-j2\pi f t_0}\\ 1 &\overset{\mathcal F}{\Longleftrightarrow} \delta (f)\\ e^{j2\pi f_0 t} &\overset{\mathcal F}{\Longleftrightarrow}\delta (f-f_0) \end{align}$$

And table:

$$\begin{array}{c|c} f(t)=\mathcal L^{-1}\left\{\mathcal F(s)\right\}&\mathcal F(s)=\mathcal L\left\{f(t)\right\}\\\hline e^{at}&\displaystyle\frac{1}{s-a} \end{array}$$

whereas others say that it is a (pole) hyperbola rather than a delta-pulse?

$$e^{at}\overset{\mathcal F}{\Longleftrightarrow}\frac{1}{s-a}$$