I have been building models with categorical data for a while now and when in this situation I basically default to using scikit-learn's LabelEncoder function to transform this data prior to building a model.

I understand the difference between OHE, LabelEncoder and DictVectorizor in terms of what they are doing to the data, but what is not clear to me is when you might choose to employ one technique over another.  

Are there certain algorithms or situations in which one has advantages/disadvantages with respect to the others?

There are some cases where LabelEncoder or DictVectorizor are useful, but these are quite limited in my opinion due to ordinality.
LabelEncoder can turn [dog,cat,dog,mouse,cat] into [1,2,1,3,2], but then the imposed ordinality means that the average of dog and mouse is cat. Still there are algorithms like decision trees and random forests that can work with categorical variables just fine and LabelEncoder can be used to store values using less disk space.
One-Hot-Encoding has the advantage that the result is binary rather than ordinal and that everything sits in an orthogonal vector space.  The disadvantage is that for high cardinality, the feature space can really blow up quickly and you start fighting with the curse of dimensionality. In these cases, I typically employ one-hot-encoding followed by PCA for dimensionality reduction.  I find that the judicious combination of one-hot plus PCA can seldom be beat by other encoding schemes.  PCA finds the linear overlap, so will naturally tend to group similar features into the same feature.
