An elliptic curve defined over the rationals gives rise to a compatible system of Galois representations. The Iwasawa invariants associated to these representations epitomize their arithmetic and Iwasawa theoretic properties. The study of these invariants is the subject of much conjecture and contemplation. For instance, according to a long-standing conjecture of R. Greenberg, the Iwasawa "mu-invariant" must vanish, subject to mild hypothesis. Overall, there is a subtle relationship between the behavior of these invariants and the p-adic Birch and Swinnerton-Dyer formula. We study the behaviour of these invariants on average, where elliptic curves over the rationals are ordered according to height. I will discuss recent results joint with Debanjana Kundu, in which we set out new directions in arithmetic statistics and Iwasawa theory.