Practical applications of the Laplace and Cauchy distributions I want to know if there are any examples of real-life applications of the Laplace and Cauchy density functions. How do they differ in their applications?
This related post, however, does not answer my question.
 A: One example is using them as robust priors for regression parameters, where Laplace prior corresponds to LASSO (Tibshirani, 1996) , but $t$-distribution, or Cauchy are other alternatives (Gelman et al, 2008).
Moreover, you can have L1 regularized regression with Laplace errors (i.e. minimizing absolute error).
Another example: Laplace noise is used in currently trendy field of differential-privacy.

Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society. Series B (Methodological), 267-288.
Gelman, A., Jakulin, A., Pittau, G.M., and Su, Y.-S. (2008). A weakly informative default prior distribution for logistic and other regression models. The Annals of Applied Statistics, 2(4), 1360-1383.
A: I would like to add an interesting case where the Cauchy distribution can arise in Biology. 
Imagine you are a shark roaming around a space $A \subset \mathbb R^3$ in search of food, which is represented mathematically by the neighborhoods $N_\epsilon(x_i)$ of several food sources $x_i$. 
Brownian motion refers to the random walk process of swimming to a new location where the change in each dimension is a normally distributed random variable. If the normal distributions have mean zero, the shark is guaranteed to find a food source eventually.
 
When food sources are scarce, the shark may abandon Brownian motion in favor of a Levy flight, in which the random walk distribution is a Cauchy distribution (or some other heavy tailed distribution). In Levy flight, the shark explores the space in a "greedy" way, sacrificing depth of the search in order to explore the space more quickly.

This is referred to as the Levy flight foraging hypothesis. The movement of sharks, swordfish, ants, albatross, humans and other species have been shown to be well-modeled by levy flight (compared to traditional Brownian motion) in a wide variety of scenarios (although the evidence is somewhat mixed). In many mathematical settings, Levy flight is demonstrably faster at blindly searching a space and is often used in robotics competitions. 
A: The Laplace distribution is also related to median linear regression models. For a model:
$$y_i=x_i^T\beta + \epsilon_i,$$
where $\epsilon_i$ are iid Laplace with location $0$ and scale $\sigma$, the maximum likelihood estimators of $\beta$ coincide with the median regression estimators
$$\hat{\beta} =\text{argmin} \sum_i\vert y_i-x_i^t\beta\vert.$$
See: https://en.wikipedia.org/wiki/Quantile_regression
The half-Cauchy prior is very popular in Bayesian hierarchical models:

Nicholas G. Polson and James G. Scott (2012). On the Half-Cauchy Prior for a Global Scale Parameter. Bayesian Analysis.

