In DnD 3.5 and Pathfinder, are there any provisions for characters wielding weapon types not on the given lists, like meat cleavers or swordcanes? Are those not allowed, do you use them as equivalent weapons (e.g. meat cleaver = dagger or shortsword, swordcane = just having a club and a shortsword), or can you tweak some stats to make them as new weapon types?
Both systems define "improvised weapons," found in the weapons rules.
Sometimes objects not crafted to be weapons nonetheless see use in combat. Because such objects are not designed for this use, any creature that uses an improvised weapon in combat is considered to be nonproficient with it and takes a –4 penalty on attack rolls made with that object. To determine the size category and appropriate damage for an improvised weapon, compare its relative size and damage potential to the weapon list to find a reasonable match.
Note there there are also some feats involving improvised weapons, such as Improvised Weapon Mastery and its prerequisites.
A meat cleaver would definitely be an improvised weapon, since it was not designed for combat but rather as a tool. I would probably consider it a hand-axe in terms of damage.
A sword-cane, however, would probably be an exotic weapon rather than an improvised one, and though there are no rules against it, since it is not defined in the rules, you would need to come up with its statistics with the approval of your DM.