How can I find out which hechsheirim are reliable and which ones aren't?
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"Reliable" is a loaded word. Let's try "accepted among conventional American Orthodox standards as we know them."
A good first place to try is Rabbi Eidlitz's kosherquest list:
It's not necessarily comprehensive, and occasionally people may nitpick with it, but it's a good first-order approximation.
In Israel, there's a great deal of politics. But here's a pointer: Rabbi Dovid Miller of the Gruss Kollel has a talk he gives to new students listing which Israeli hechshers are "mehadrin", and thus recommended. The audio is available on YUTorah.org, and I believe the accompanying handout sheet (PDF) is floating around the internet somewhere. I'm told that Michlalah students are given a similar (if not identical) list.
If it's a local hechsher (e.g. a restaurant certified by the Rabbinical Association of Anytown, USA), contact a local rabbi.
Generally, in the US today, a hechsher that will allow "non-Glatt" beef, such as Hebrew National, is not considered up to the standards used by most Orthodox Jews in America today. Similarly, if it's a meat restaurant, it's open on Shabbos, and it's located someplace where the mashgiach couldn't reach it on Shabbos, that's usually a bad sign.
Otherwise, you can email the webmaster at kashrut.com, who stays in touch with lots of rabbis in the business. The OU does not publicize which other hechshers it accepts, but I believe Chicago's cRc might.
Beyond the US or Israel, try contacting a local rabbi. Different countries can have different standards, and there's an argument to follow local standards (as long as they're within reasonable Halachic boundaries) when visiting. The Chavos Yair wrote of a town in Alsace where certain dried fruit required a hechsher, while it didn't elsewhere. The Kitzur Shulchan Aruch says that some cities required a hechsher on plain butter, and some didn't, but to follow the practice of where you're located at this moment.
When evaluating the status of liquors, especially European ones, I've found the London Beth Din's directory to be quite comprehensive.
For Israel kosher news (with a mehadrin, charedi bent), see:
For updates from the Chief Rabbinate of Israel: