> **Theorem** (Xin Wan): If $E$ is an elliptic curve of square-free conductor $N$, and $p \ge 3$ is a prime such that $p \nmid N$ and $a_p(E) = 0$, then Kobayashi's $\pm$  Iwasawa main conjectures are true for $E$ (for both choices of sign).

The proof is in [this paper][1]. As the introduction to the paper notes, the assumption of $N$ being squarefree could potentially be relaxed, but I don't know if anyone has pushed this further.

Supersingular elliptic curves with $a_p \ne 0$ can only occur for $p = 2$ or $p = 3$. The case $p = 3$ has been proved by Sprung in [this paper][2], using an extension of Wan's methods.

As for higher weight modular forms, Lei formulated a main conjecture for $a_p = 0$ in his PhD thesis as you apparently know, and there is a more general formulation covering the $a_p \ne 0$ case due to [Lei, Zerbes and myself][3] (2010). This conjecture has been proved under some technical assumptions by Wan in [this paper][4].

So the case of good ordinary and good supersingular reduction is getting to be quite well understood now. On the other hand, for bad (additive) reduction we know <del>almost nothing</del> rather less -- in particular, in this case there is (AFAIK) no satisfactory way to formulate a main conjecture in the form "p-adic L-function = char. ideal of Selmer group". There are other approaches to formulating a main conjecture, e.g. Kato's formulation (relating the size of an $H^2$ to the index of a special element in an $H^1$), and there are some partial results known towards these; in particular, one inclusion in Kato's conjecture is known if the mod p Galois image is large.

  [1]: http://www.mcm.ac.cn/faculty/wx/201609/W020161128334766904862.pdf
  [2]: https://arxiv.org/abs/1610.10017
  [3]: https://projecteuclid.org/euclid.ajm/1298989628
  [4]: https://arxiv.org/pdf/1607.07729.pdf