Determining the rarest sword is a challenging task due to the limited availability of certain swords, the diverse range of historical periods and cultures, and the difficulty in tracking and identifying rare specimens. However, there are several swords that are considered extremely rare and hold significant historical and cultural value. Here are a few examples:

1. Honjo Masamune: The Honjo Masamune is a legendary Japanese sword believed to have been crafted by the renowned swordsmith Goro Nyudo Masamune during the 13th century. It was a prized possession of the Tokugawa shogunate and is regarded as one of Japan's national treasures. The current whereabouts of the Honjo Masamune are unknown, adding to its rarity and mystique.
2. Seven-Branched Sword: The Seven-Branched Sword, also known as the Shichishitō, is a unique ceremonial sword from ancient Japan. It is an important artifact with historical and religious significance. The sword is believed to have been a symbol of authority and power and is considered one of the oldest surviving swords in Japan.
3. Sword of Goujian: The Sword of Goujian is an ancient Chinese sword discovered in the tomb of King Goujian of the Yue State, dating back to the Spring and Autumn period (771-476 BCE). The sword is renowned for its exceptional preservation, intricate design, and advanced craftsmanship for its time.
4. Szczerbiec: Szczerbiec is a ceremonial sword associated with Polish kings. It is a symbol of Polish monarchs and has been used in coronation ceremonies. The sword's rarity and significance lie in its historical importance as an emblem of Polish monarchy and sovereignty.

These swords are highly regarded for their historical and cultural value, and their scarcity contributes to their rarity. However, it's important to note that many rare and significant swords exist beyond these examples, and the rarity of a sword can vary based on its historical context, craftsmanship, cultural importance, and individual perspectives.