After a crime has been perpetrated, but before the criminal has been apprehended, is it permissible to pray that the perpetrator wasn’t Jewish or does this fall under the category of tefillat shav? Thank you.

Shalom, Thank you for your question. The Mishna in tractate Brachot (chapter 9, mishna 3) states that "He who supplicates for that which has already taken place, offers a prayer in vain. If [for instance] a person whose wife is pregnant, prays, "God grant that my wife bring forth a male child," or if a person on the road hears loud lamentations in the town and prays, "God grant that it may not concern any of my family." Such as these are prayers in vain." As you point out, praying that the perpetrator of a crime was not a Jew would fall into this category, and as such be forbidden. When people say, after hearing that some terrible crime has been committed, "I just pray that it wasn't a Jew who did it", they are probably not really offering up a prayer to G-d, but rather expressing their personal hope that a Jew, who should hopefully have raised themselves above such levels of base sin through keeping the Laws of the Torah that Hashem gave us, would not have sunk to such a level of crime. It is certainly not forbidden to hope that it was not their family in a given tragedy, or that one's baby be a boy or a girl. (I have also been known to hope that I passed my exams, have good blood pressure, and bought everything I was supposed to from the shop). What is forbidden is to pray to Hashem for such a retroactive outcome. The reason this is forbidden is that the outcome is already fixed, and one should turn their prayers to the future, and not to expect that Hashem will do you a miracle against the laws of nature, by "turning back the clock" and changing what has already been fixed in the world. Hoping that a certain thing happened, on the other hand, is permitted. This is an expression of one's inner desires, that it is often (though not always) good and healthy to express. When I hope that the dinner I cooked is tasty, I am expressing an inner desire to treat my guests well, which is a good thought to strengthen in myself. And so to, the desire and hope that a crime was not committed by a Jew is a good thought that one should strengthen in themselves. Then, if unfortunately it turned out that a Jew did in fact commit the crime, hopefully, the disappointment will be the catalyst to work harder to uplift one's community so such things will never occur again. May we be blessed with only good news.