A will always be
a knight
Here's one possible solution for the questions and identities of the three:
$S$ is
the set of the two possibilities where B is normal. In other words, it is equivalent to asking "is B normal?"
When asked if $S$ contains the correct possibility
A said yes. This could mean A is normal, or that A is a knight and B is normal, or that A is a knave and B is the knight.
When asked about C, B said
C is a knight.
Using this, we can determine the identities of A, B, and C:
Neither the knight nor the knave would claim the other is a knight, so A is not normal. Based on the response to the first question we know B is either normal or the knight, but the knight would not claim anyone else was the knight so B must be normal. So A is the knight, B is normal, and C is the knave.
Here's another possibility:
$S$ is
the set {(Knight, Normal, Knave), (Knave, Knight, Normal)}.
When asked if $S$ contains the correct possibility:
A said no. This could mean A is normal, or that A is a knight and the first possibility in the set swaps the normal and knave, or that A is a knave and the second possibility in the set is correct.
When asked about C, B said
C is a knight.
Using this, we can determine the identities of A, B, and C:
Neither the knight nor the knave would claim the other is a knight, so A is not normal. A can't be a knave either, because then B would be a knight claiming a normal was a knight. So A is the knight, B is the knave, and C is the normal.