The depictions in blogs and the "alternative" press make it sound like the 4's trial was a complete perversion of justice. I rather hope that is the case. Because their fate will change only if the transcript shows blatant and obvious bias or error on the part of Justice McLaughlin.
I don't practice criminal law, but here's a quick and dirty summary of how this works: In appealing the verdict and sentence they received, the 4 must point to specific flaws in the process to date. Commonly, appeals rest on erroneous evidentiary rulings (as seen on Law and Order -- you argue that important information was improperly excluded or included at trial) or flawed instructions to the jury when it came time for them to deliberate. If the appellate court agrees that there was a problem, it will then rule on whether the error was "harmless" (not likely to affect the outcome of the trial), and if not, order a new trial (or some other remedy). Should the 4 strike out in the New York State appellate courts, they could then seek a writ of habeas corpus in federal court (i.e. argue that their state trial was so unfair as to be unconstitutional).
Bloggers say (can't find the reference at this moment) that the 4's lawyers wanted to explain to the jury about their personal acquaintance with Sakia Gunn, and tell the jury Ms. Gunn's story, but that evidence was excluded. Again, this is not my field of law, I didn't observe the trial, and I don't know if that argument has as much legal force as it does emotional force, but it's among the possible avenues of appeal. It seems likely that Judge McLaughlin would have excluded a whole host of lines of questioning/types of evidence related to the 4's subjective experience of their situation, and that in turn would make an evidentiary appeal easier, or at least provide more pitches for the appellate defense attorney to swing at.
It would really help the appeal project if the judge had made statements reflecting bias or malice toward the defendants in open court. Some depictions of the 4's trial sound like that may have happened. (e.g. Imani Henry: "McLaughlin stated throughout the trial that he had no sympathy for these women.") On the other hand, if the judge managed to express his hatred for these defendants subtly, nonverbally, and/or on an even keel with the way he's treated *other* defendants, they will have no remedy...a certain amount of disrespect is standard in the criminal justice system.
As for jury instructions -- Justice McLaughlin happens to have been called out recently by a federal judge for having a history of "unorthodox and incorrect jury instructions" -- that was in a habeas corpus proceeding where the federal court overturned the conviction in a drug trial McLaughlin had presided over. That's rather a good sign for those of us hoping the 4's conviction will get overturned on legal grounds. (FWIW, Justice McLaughlin has also received public attention in this case and in at least one other for being a hard-ass toward folks he perceives to be trying to evade jury duty.)
Finally, several friends and colleagues of mine have read or heard a quick synopsis of the Jersey 4 story and responded that they must have lacked for good legal representation. I thought this too, when I first started reading about it; I think it's wrapped up in a sense of denial among people with plenty of race/class privilege about how dysfunctional our justice system is for poor people of color. Anyway, the New York Times article and some blog accounts note that Renata Hill, at least, was represented at the trial level by Susan Tipograph. Tipograph is a very highly regarded criminal defense attorney specializing in "political prosecutions" -- she's involved with the National Lawyers' Guild and has represented David Gilbert, Lynne Stewart, and a lot of other famous folks. I don't know if all four women had access to high-quality representation at the trial level, nor do I know if their legal needs are being well met now (as of July they were begging for help on this front), but at least one experienced and progressively-committed lawyer has definitely been on the case.