For example, almost all one-relator groups with more than two generators are residually finite. The people who proved this actually showed that something like 98% of one-relator groups with more than two generators are residually finite. Now, it is "well-known" (for a suitable value of "well") that not all one-relator groups are residually finite. Famously, the group is not residually finite (because it doesn't have a weaker property: it isn't Hopfian), and if you know enought about all this stuff you can quite easily see that is not residually finite. So, you have a result which says "almost all one-relator groups with more than two generators are residually finite, but some are not". Which is, holistically speaking, very nice.
Their proof of the 98% was quite neat. A one-relator group is a group of the form where is a word over the alphabet . This word defines a walk in -dimensional space (if you assume, without loss of generality, that every letter in appears in ), and so the authors proved that if you take a random walk it "almost always" has a specific property which implies that the corresponding group is residually finite (they proved that words which define a certain type of walk yield groups which are "ascending HNN-extensions of free groups"). The paper is by Mark Sapir and Iva Spakulova.
(A group is residually finite if for any there exists a homomorphism where is some finite group, such that ( denotes the itentity). For instance, is residually finite (here, denotes the identity), as taking an arbitrary integer then it is mapped to modulo . It is a rather strong finiteness condition.)