An open question in coding theory asks whether or not MRD codes with the rank metric are dense as the field size tends to infinity. In this talk, I will briefly survey this problem and its connections with the theory of spectrum-free matrices and semifields. I will then describe a new combinatorial method to obtain upper and lower bounds for the number of codes of prescribed parameters, based on the interpretation of optimal codes as the common complements of a family of linear spaces. In particular, I will answer the above question in the negative, showing that MRD codes are almost always (very) sparse as the field size grows. The approach offers an explanation for the b divergence in the behaviour of MDS and MRD codes with respect to density properties. I will also present partial results on the sparseness of MRD codes as their column length tends to infinity. The new results in this talk are joint work with A. Gruica.