smc

Consider a circle whose center lies on the positive $y$-axis and which passes through the origin. As the radius of this circle becomes arbitrarily large, its curvature near the $x$-axis becomes almost flat, and so it can intersect the curve $y=\sin x$ arbitrarily many times (since the $x$-axis itself intersects the curve infinitely many times). Hence, in particular, we can choose a radius sufficiently large that the circle intersects the curve at $\boxed{\text{more than }16\text{ points}}$.