A creature that possesses a base attack bonus of +1 or more "may draw a weapon as a free action combined with a regular move" (see here). Although the game never says it outright, this reader and everyone he's played with has always assumed that a regular move means "[t]he simplest move action[, i.e.] moving your speed" (see here).
Further, "If you have the Two-Weapon Fighting feat, you can draw two light or one-handed weapons in the time it would normally take you to draw one," and that includes during a regular move (see Note 3 here).
Thus, unless the GM rules otherwise, a weapon (or sometimes two weapons) can be drawn when a creature that possesses a base attack bonus of +1 or more takes a move action to move up to its speed. Other move actions typically don't allow the simultaneous drawing of a weapon.
Also, sheathing a weapon shouldn't be conflated with drawing a weapon. It's actually much more difficult to sheathe a weapon quickly than to draw one. (See this question.) The two actions are entirely separate and unique actions with their own costs. A creature that possesses a base attack bonus of +1 or higher that takes a move action to move typically can't simultaneously sheathe its weapon, for example.