Runners on second (trailing runner) and third (lead runner). Batter hits fair ball. Trailing runner advances to third but the lead runner doesn't advance, leaving both runners standing on third base. Fielder tags both runners. Is the lead runner out? Is the trailing runner out?

Answer: Because the runner initially on second base was not forced to advance, the lead (preceding) runner is safe and the trailing (following) runner is out. Note: had the runner on second been forced to advance, meaning a runner on first, then the lead runner is out and the trailing runner is safe.

Under Official Baseball Rules Rule 7.03
(a) Two runners may not occupy a base, but if, while the ball is alive, two runners are touching a base, the following runner shall be out when tagged and the preceding runner is entitled to the base, unless Rule 7.03(b) applies.
(b) If a runner is forced to advance by reason of the batter becoming a runner and two runners are touching a base to which the following runner is forced, the following runner is entitled to the base and the preceding runner shall be out when tagged or when a fielder possesses the ball and touches the base to which such preceding runner is forced.

If unsure which runner to tag, tag each of them and let the umpire decide which one is out.