As a matter of progression most anthropologists will claim that the spear was the first constructed weapon (the first weapon as we understand the word would have been a sharp rock or stick), as man would have figured out how to sharpen a stick into making it useful.
The next weapon (once man figured out how to attach stone to a stick) would be the axe.
So as far as melee weapons go spears and AXES would be the most common melee weapons. You might make a case for a hammer since early axes were bludgeoning weapons, and thus it was really just attaching a heavy object to the end of a stick.
Swords were developed much later in history and as such primitive cultures in Fallout will not have achieved the level of technology to make one, and advanced cultures aren't going to give a crap because they are producing black powder weapons.
As a matter of utility knives would be rather common. I emphasize utility because knives are not going to be as common for weapons because in a fight a knife generally loses to a spear, and thus you will see knives as hunting tools, carving tools, and as a general tool used to pry open things since most people aren't going to have access to a crowbar.
Machete's might be common for jungle mexico, but not the heartland of the US. And as far as patchwork jobs we would be more inclined to find a metal pole turned into a spear or a metal scrap turned into an axe.
Now I would definitely expect slings and arrows to exist, and as far as range weapons go those would be far preferable to the makeshift or relic throwing knives and darts that would be found. A game dart just isn't all that lethal, and most makeshift throwing knives would not be balanced very well.
Now as far as patchwork jobs that I would go for: thermite for anti-power armor and white phosphorus for anti-personnel. Thermite is a mixture of aluminum powder and rust (preferably black rust, but red rust works), and it burns at several thousand degrees C. Now a scientifically minded individual might know the proper chemical ratio of 1-3 aluminum to iron oxide. And I would think it very easy to scrounge rust off old steel nor do I imagine finding aluminum to grind down would be hard to find either (especially what with people not building large scale machinery anymore). Thus incendiary weapons could become a reality. *Note* Thermite has a high ignition temperature, thus something else is needed to jump start the reaction. Magnesium or white phosphorous would both be reasonable assumptions for starting methods. And as far as WP goes, according to online, its properties have been known since prior to the 20th century, and thus we could probably pretty reasonably expect it to have continued use in the Fallout world.
If its a home made gas grenade, then I would look out for household cleaners and cleansers. Assuming that people hadn't raided those things already (I don't imagine that those would be raided by scroungers since they aren't useful to people post-war barring the occassional hygiene minded person). Mixing ammonia and bleach produces chlorine gas which as we all know is an effective chemical warfare agent
But in all honesty without a machine shop it would be hard to produce good makeshift weapons (I mean barring primitive stuff). After the war I would assume just about every hardware store within driving distance would have been raided for supplies. So I imagine by now only the Brotherhood or large towns might have consolidated anything resembling a machine shop. But see once you have the proper tools I would start making brass jackets for bullets and work towards the creation of boiler to carry out the bessemer process on old steel to help remove enough impurities to net useful metal (not sure what I would use since high grade materials would be hard to come by, and I'm not holding out for a molybdenum water cooled crucible to just be lying around some place for the taking).
