You should ideally NOT play scrabble on Shabbos. (Just google “playing scrabble on Shabbos.”)
Chazal made a decree prohibiting the playing of games where writing is part of its regular activity [Chayei Adam 38:11; see, however, Bris Olom Kosev 13 page 45 who does not know where the Chayei Adam got this from). Therefore, one may not play Scrabble or any other game where people usually keep score. [Refer to Tiltulei Shabbos page 25 footnote 24, Igros Moshe O.C. 5:22:14.] Poskim permit playing chess on Shabbos even though some people write down their moves. This is permitted because most people do not write down their moves.
If the game involves placing the pieces in a frame to hold them together then it would be forbidden to play with
one does not place the pieces in a frame to hold them in place.
Tiltulei Shabbos page 25, Nishmas Shabbos 6:419.]
[Shalmei Yehuda 5:footnote 5.]
Rabbi Pinchas Scheinberg zatzal permitted playing Scrabble on Shabbos as long as you are EXTREMELEY careful not to c”v write down the score.
In truth, however, you should spend the Shabbos LEARNING TORAH!