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. 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, 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 (2010). This conjecture has been proved under some technical assumptions by Wan in this paper.

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 almost nothing 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.

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. 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, 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 (2010). This conjecture has been proved under some technical assumptions by Wan in this paper.

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 almost nothing

In the bad additive reduction case, there is (AFAIK) no satisfactory way to define a p-adic L-function in this case, hence no way to formulate a main conjecture in the form "p-adic L-function = char. ideal of torsion Selmer group". However, are other approaches to formulating a main conjecture: in particular, there is Kato's formulation from his 2004 Asterisque paper, relating the size of an $H^2$ to the index of a special element in an $H^1$. Under a modest "big Galois image" assumption, Kato proves one inclusion in this formulation, and I learned from Olivier's comment that he and Wan have proved the other inclusion under similar hypotheses, so Kato's main conjecture is now a theorem in this case as well.

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. 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, 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 (2010). This conjecture has been proved under some technical assumptions by Wan in this paper.

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 almost nothing.

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. 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, 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 (2010). This conjecture has been proved under some technical assumptions by Wan in this paper.

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 almost nothing 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.