At first glance, it seems that the weapon selection in 5e D&D is quite limited, but with a little knowledge of the system, you can largely expose that template that builds those weapon, and from there, well, the local blacksmith is the only limitation to the weapons you can find!

You can divide your damage die into smaller dice that equal the same total. For example, a d12 can become 2d6 or 3d4.

To create a weapon, you take a d6, go through five steps to determine the final damage and properties of the weapon.

Not-so-common Weapons.

Taking the formula, we can now make all the missing weapons of the game world. While these weapons should not necessarily be assumed to exist and are not starting gear, perhaps you can craft them, convince the local blacksmith to give your idea a go, or find these as loot early in your adventures - the strongest of these weapons tend to still fall below the threshold of Uncommon, but some will be more powerful for some builds than common starting gear weapons.

Below are some results of the template. Note that all weapons (even if they are created by the template) only count under existing weapon proficiency at the discretion of your DM.