When given samples of a discrete random variable, the entropy of the distribution may be estimated by $- \sum \hat{P_i} \log{\hat{P_i}}$, where $\hat{P_i}$ is the sample estimate of the frequency of the $i$th value. (this is up to a constant determined by the base of the log.) This estimate should not be applied to observations from a continuous distribution, at least naively, because it would yield a value which depends only on the sample size.
Beirlant et al describe a number of approaches for the continuous problem, including estimates based on empirical CDF, nearest neighbor distances and the $m$-spacing estimate, which is given by $$\frac{1}{n}\sum_{i=1}^{n-m}\log{(\frac{n}{m}(X_{(i+m)} - X_{(i)}))}$$, where $X_{(i)}$ is the $i$th order statistic of the sample, and $m$ varies in a certain way with $n$. It is not clear how this estimate is to be computed in the presence of ties, i.e. it does not appear to be applicable to discrete distributions. (a naive correction for ties (drop terms which have $\log{0}$) appears to give an estimator which does not depend on the relative frequency of the classes, only their values, which seems wrong.)
The question: is there an 'all-purpose' estimator which can deal with both discrete and continuous (or even mixed) distributions?
