Being a basic Wild Words item, the primary use for Primordial Shrieks is to convert them up to Maniac's Prayers.
You can use Primordial Shrieks in batches of 1, 50, and 500 by clicking on Primordial Shrieks in your inventory, which you can find on the "Myself" tab.
Here are the choices you currently have:
There is also a list of other uses at the Fallen London Wikia, but they are few and far between.
As with most things in Fallen London, you could of course pawn your Primordial Shrieks at the Bazaar, where they sell for 2 pence each.
These acquaintances are unlocked in several places, and they have storylets for increasing your friendship with them, usually via buying them something or doing a favour.
The Repentant Forger - If you end up with your Suspicion above 8, you'll be thrown in New Newgate again, and can meet him via an Opportunity Card. Befriending him will allow you to meet him in Veilgarden, where he can help you with increasing your Fascinating quality.
The Regretful Soldier - You can met The Regretful Soldier via an opportunity card while in Watchmaker's Hill, as long as your Dangerous quality is above 45. If you talk him down and prevent him from fighting (which involves having a Persuasive quality of around 45), he'll become your friend.
The Sardonic Music-Hall Singer - A character within the 'Missing Person: Absconding Devil' series of storylets in Ladybones Road (unlocked at Watchful 12), as well as the 'Bag A Legend' ambition. She also appears in an Opportunity Card that is unlocked with a Persuasive quality of at least 89. Each of these will let you befriend her.
The Wry Functionary - If you attend the Ambassador's Ball in Veilgarden (Persuasive quality of 56 needed) and get the 'The Darling of the Ambassador's Ball' quality as a result, you'll be able to correspond with him via another storylet in Veilgarden. On a rare success, you'll gain him as an Acquaintance, and you can meet with him in the Shuttered Palace once you gain access to it.
Best Answer
It's not hugely helpful, but if you have 15 Admired you can laugh off Scandal instead of needing a friend to redirect it onto.
Edit from 2013: Admired has been renamed to Making Waves, and is now critical to making promoted professions available.