As fine as frog's hair
Extremely fine, that is, delicate and slender.
This is an American simile and dates back to the mid 19th century. C. Davis's Diary of 1865 has this entry:
"I have a better flow of spirits this morning, and, in fact, feel as fine as frog's hair, as Potso used to say."
The allusion to the hairs on a frog clearly points us to the 'slender, narrow', meaning of the phrase. Just as clearly, frogs don't have hair, and the ironic reference to it is intended to highlight the effect. This is similar to the British simile 'as rare as rocking-horse shit' that is, nonexistent. The British also have a related vulgarism, which takes the implied smallness a step further - 'as small as the hairs on a gnat's bollock'.
The citation above plays with the meaning of the expression by using 'fine' to mean 'excellent - in high spirits'.
There is a lesser-known variant from the southern states - 'as slippery as frog hair'. This is used to denote money, especially that which is newly acquired; for example, from Time, February 1974:
"Disturbingly, many of the plaque owners were contractors or architects who stood to benefit from making political contributions - frog hair, as such funds are known... because, as old Sooners [Settlers who jumped the gun and arrived too soon to a claim] say, new money feels 'as slippery as frog's hair'."
See other 'as x as y similes'.