What are the connotations of "Lo Sa'aneh ... Eid Shav" (as opposed to "Eid Shaker" in the first Luchos)?
Ramban: "Eid Shav" means even a vain testimony, which will not cause the relevant litigant a loss in Beis-Din. 1
Meshech Chachmah: The Rashbam says that it is Sheker that will become known, e.g. to say that Plonis' husband died [in order to marry her]. The Torah believed one witness about this, but Chachamim do not let him marry her, so it is Shav.
Ha'Emek Davar: If only Reuven saw Ploni transgress, he may tell Ploni's Rebbi, so he will hate him, only if the Rebbi believes Reuven as if he were two witnesses (Pesachim 113b). If not, he is Ed Shav. If he tells Shimon, who believes him like two, and Ploni's Rebbi believes Shimon like two, Shimon may not tell the Rebbi 'I saw it', for if Reuven would tell the Rebbi, he would be an Ed Shav. .
Ramban: Such as where the witness testifies that Reuven undertook to give Shimon a Manah without making a Kinyan. See also R. Chavel's footnotes.
Sotah 2a learns from "Lo Yakum Ed Echad b'Ish" (19:15) that elsewhere, "Ed" is two witnesses. Why don't we learn from a Binyan Av, that everywhere it is one? Also, we could learn from "v'Ed Echad Lo Ya'aneh" (Bamidbar 35:30), and "Lo Sa'aneh Ed Shav"
Refer to Bamidbar 5:13:153:1,2 and the note there.