Let $A=(-1,0),B=(1,0), C$ be points in $\mathbb{R}^2.$ What is the locus of points $\{s\in\mathbb{R}^2:C$ lies on the angle bisector of $AsB$}? Let $A=(-1,0),\ B=(1,0),\ C$ be points in $\mathbb{R}^2.$ What (shape(s)) is the locus of points $L(C)=\{s\in\mathbb{R}^2:C$ lies on the angle bisector of $AsB$ } ?
Obviously the set $L$ depends on where $C$ is, which is why I've written $L$ as a function of $C.$
If $C$ is the midpoint of $AB$, or on the perpendicular bisector of $AB$, it is not clear to me that $L= \{$ the perpendicular bisector of $AB\},$ because maybe $L$ contains more points. [Edit: Karan Elangovan has answered this part of the question.]

The locus you are asking about is a subset of a cubic curve called an Apollonian strophoid (or strophoid of Apollonius).
The curve is the set of points $S$ for which either the internal or bisector of the angle $\angle ASB$ passes through $C$.  (The external bisector is the line through $S$ perpendicular to the internal bisector.)  As the strophoid $\Sigma$ comes in from infinity, $C$ is on the internal bisector but, as $\Sigma$ goes past $A$, $C$ is on the external bisector until $\Sigma$ passes $B$ after which $C$ is back on the internal bisector all the way to infinity.  So your locus is actually the strophoid minus the section between $A$ and $B$.
A GeoGebra applet that demonstrates the strophoid can be found here.  The strophoid is blue, the external bisector is green. You can move $S$ and $C$.  (Unfortunately the $S$ will jump around if you modify any of the points.  This is annoying but harmless.)  The strophoid is built using mathlove's equation.

There are two special cases where the cubic degenerates to a line $\ell$ and a circle $c$.   Both have been touched upon in other answers and comments for this question.

*

*when $C$ is on the perpendicular bisector of $A$ and $B$, line $\ell$ is the perpendicular bisector and $c$ is the circumcircle of $A,B,C$.  For points $S$ on $c$ it is the external bisector that goes through $C$.  (Because, if $M$ is the midpoint of $A,B$ then the triangle $\triangle ECS$ is a right angle.


*when $C$ is on the line $AB$, line $\ell$ is the line $AB$ and $c$ is an Apollonian circle of the segment $AB$.
For more about strophoids see the following references.
Gibert, Apollonian strophoids (part of a large catalog of so-called Triangle cubics)
mathcurve.com (Strophoid page)
Stachel, Strophoids, A Family Of Cubic Curves With Remarkable Properties (see Theorem 4).
