The singularity of something is a quality that's unusual or unique. You might note the singularity of a Hollywood movie that features a female lead role and no romance, for example.
You can use the noun singularity when something is literally "one of a kind," like your grandmother's strange macrame wall art, or to mean exceptional or striking, like your nephew's bright red hair. He stands out among the other kindergartners, so his hair has a singularity, though he's not the only red-headed kid in the world. The Latin root is singularis, which means "single or solitary."