With 1 out, runner on the 3rd base properly tags up and crosses the home plate before the runner on 2nd base who does not tag up properly going to 3rd and is called out resulting in the 3rd out of the inning. Does the run score?
Yes, the run counts. Run scoring is covered by rule 5.08(a)
5.08 How a Team Scores
(a) One run shall be scored each time a runner legally advances to and touches first, second, third and home base before three men are put out to end the inning.
EXCEPTION: A run is not scored if the runner advances to home base during a play in which the third out is made (1) by the batter-runner before he touches first base; (2) by any runner being forced out; or (3) by a preceding runner who is declared out because he failed to touch one of the bases.
There are three exceptions listed, but none of them apply in this case since the out at third base is not a force.
The MLB rulebook also contains an Approved Ruling scenario that is very similar.
APPROVED RULING: One out, Jones on third, Smith on first, and Brown flies out to right field. Two outs. Jones tags up and scores after the catch. Smith attempted to return to first but the right fielder’s throw beat him to the base. Three outs. But Jones scored before the throw to catch Smith reached first base, hence Jones’ run counts. It was not a force play.