Twice. The subsolar point occurs on the equator during the two equinoxes.
Subsolar point via kwout
On these dates, the equator is lined up with the ecliptic plane and the poles are in line with the circle of illumination. During the summer solstice, the subsolar point moves to the Tropic of Cancer (23.5° N) because at this time the North Pole is inclined 23.5° toward the Sun.
Subsolar point is the location on the Earth where the Sun is directly overhead at solar noon. The subsolar point is an area where the sun's rays shine perpendicular to the Earth's surface—a right angle.
Only during an equinox is the Earth's 23.5-degree axis not tilting toward or away from the sun: the center of the sun is in the same plane as the Equator.
Most planets experience equinoxes. On the gas giant Saturn, equinoxes are particularly dramatic. Saturn's spectacular ring system extends along the same plane as the planet's equator. Although the rings extend thousands of kilometers into space, they are actually very thin, only about a kilometer wide. During Saturn's equinoxes, the rings (and Saturn's equator) line up perfectly with the sun. Photos taken from the solar perspective reveal the rings as a razor-thin line.
For more info, go to Nationalgeographic.org and Physicalgeography.net.