Here is a very nice, practical use: an algorithm for computing graph connectivity: http://www.cse.cuhk.edu.hk/~chi/papers/conn.pdf To compute the s->t connectivity of a graph, the authors give an algorithm that assigns random vectors with entries drawn from a finite field to the out edges from s, then construct similar vectors for all of the edges in the graph by taking random linear combinations, and finally discover the connectivity by computing the rank of the resulting vectors assigned to the in-edges of t.

Here is a very nice, practical use: an algorithm for computing graph connectivity (from FOCS2011). To compute the s->t connectivity of a graph, the authors give an algorithm that assigns random vectors with entries drawn from a finite field to the out edges from s, then construct similar vectors for all of the edges in the graph by taking random linear combinations, and finally discover the connectivity by computing the rank of the resulting vectors assigned to the in-edges of t.