AFAICT none of the Syrian rebels that tried to form governments of their own have tried to apply to join the ICC, unlike the PA.
On 1 January 2015, the Government of Palestine ("Palestine") lodged a declaration under article 12(3) of the Rome Statute of the International Criminal Court (“Rome Statute” and “Court” or “ICC”) accepting the jurisdiction of the Court over alleged crimes committed "in the occupied Palestinian territory, including East Jerusalem, since June 13, 2014". On 2 January 2015, Palestine acceded to the Rome Statute by depositing its instrument of accession with the UN Secretary-General. The Rome Statute entered into force for Palestine on 1 April 2015.
N.B. there was an editorial in WaPo in 2014 that suggested the opposition groups in Syria do that, but AFAICT they didn't try.
I'm not sure we've seen the last chapter of this, but it's perhaps worth adding that despite the ICC not having/claiming jurisdiction over Syria proper, there was actually an attempt to make them take a case against Assad (and Iran too!), based on the crime of deportation, because the some of the victims fled to Jordan, which is an ICC member.
The request includes evidence of Syrian victims forced to flee into Jordan due to attacks and intimidation by the Syrian government and Iran-backed militia groups. It is being brought by the US-based Iran Human Rights Documentation Center in conjunction with Haydee Dijkstal, a UK barrister. [...]
Syria is not a party to the ICC’s Rome Statute, but it is argued the ICC has jurisdiction because the victims fled into Jordan, which is a state party. [...]
Article 7(1)(d) of the Rome Statute grants the court jurisdiction over the crime against humanity of “deportation or forcible transfer of population”, meaning the “forced displacement of the persons concerned by expulsion or other coercive acts from the area in which they are lawfully present, without grounds permitted under international law”.
In a previous case in 2018, the ICC found it had jurisdiction over the Rohingya people when they were forced to flee into Bangladesh refugee camps from Myanmar. Bangladesh, unlike Myanmar, is a party to the ICC.
However, AFAICT, neither that [attempted] case nor the similar one involving the Rohingya has made much headway (as reported in the summer of 2023).
The chief prosecutor acknowledged the frustration felt at the speed at which the court issued an arrest warrant for Russian President Vladimir Putin for an alleged scheme to deport Ukrainian children to Russia. Meanwhile, the Rohingya have been waiting six years and no such action has been taken against the Myanmar military leaders who ordered the attacks.
“The big difference is that we have access to Ukraine, we don’t have access to Myanmar,” Khan said.
Although CNN says the junta 'ordered' those attacks, the junta in fact claims that they didn't, and made some cover-up efforts ("the military has insisted the operation was a legitimate counter-terrorism campaign sparked by attacks by Muslim militants, not a planned program of ethnic cleansing"), so I guess that case is a bit less straighforward than CNN makes it to be, legally speaking.
OTOH
While Myanmar is not an ICC member state, the National Unity Government (NUG) — the body representing Myanmar’s democratically-elected leaders — lodged a 12(3) declaration with the court’s registrar on July 17, 2021, accepting the jurisdiction of the court for international crimes committed on Myanmar territory since July 1, 2002, and into the future.
So, there's an additional argument there that the ICC has jurisdiction, in that [Myanmar] case, even though the ICC accepted it before that (NUG) declaration.