Cryptids
Cryptids are Turing Machines whose behavior (when started on a blank tape) can be described completely by a relatively simple mathematical rule, but where that rule falls into a class of unsolved (and presumed hard) mathematical problems. This definition is somewhat subjective (What counts as a simple rule? What counts as a hard problem?). In practice, most currently known small Cryptids have Collatzlike behavior. In other words, the halting problem from blank tape of cryptids is mathematicallyhard.
If there exists a Cryptid with n states and m symbols, then BB(n, m) cannot be solved without solving this hard math problem.
The name Cryptid was proposed by Shawn Ligocki in an Oct 2023 blog post announcing the discovery of Bigfoot.
Cryptids at the Edge
This is a list of Minimal Cryptids (Cryptids in a class with no strictly smaller known Cryptid). All of these Cryptids were "discovered in the wild" rather than "constructed".
Name  BB domain  Machine  Announcement  Date  Discoverer  Note 

Bigfoot  BB(3,3)  1RB2RA1LC_2LC1RB2RB_2LA1LA 
BB(3, 3) is hard  Nov 2023  Shawn Ligocki  
Hydra  BB(2,5)  1RB3RB3LA1RA_2LA3RA4LB0LB0LA 
BB(2, 5) is hard  May 2024  Daniel Yuan  
BB(2,5)  1RB3RB3LA1RA_2LA3RA4LB0LB1LB 
A Bonus Cryptid  May 2024  Daniel Yuan  
Antihydra  BB(6)  1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_0RA 
Discord message  June 2024  @mxdys , shown to be a Cryptid by @racheline . 
Same as Hydra but starting iteration from 8 instead of 3 and with termination condition

Larger Cryptids
A more complete list of all known Cryptids over a wider range of states and symbols. These Cryptds were all "constructed" rather than "discovered".
Name  BB domain  Machine  Announcement  Date  Discoverer  Note 

RH  BB(744)  https://github.com/sorear/metamathturingmachines/blob/master/riemannmatiyasevichaaronson.nql  2016  Matiyasevich and O’Rear  The machine halts if and only if Riemann Hypothesis is false.  
Goldbach  BB(27)  https://gist.github.com/anonymous/a64213f391339236c2fe31f8749a0df6(unverified)  2016  anonymous  The machine halts if and only if Golbach's conjecture is false. To the best of our knowledge this construction has not been independently verified.  
Erdős  BB(5,4) and
BB(15) 
arxiv preprint  Jul 2021  Tristan Stérin (@cosmo ) and Damien Woods 
The machine halts if and only if the following conjecture by Erdős is false: "For all n > 8, there is at least one 2 in the base3 representation of 2^n"  
Weak Collatz  BB(124) and BB(43,4)  https://docs.bbchallenge.org/other/weak_Collatz_conjecture_124_2.txt (unverified)
https://docs.bbchallenge.org/other/weak_Collatz_conjecture_43_4.txt (unverified) 
Jul 2021  Tristan Stérin  The machine halts if and only if the "weak Collatz conjecture" is false. The weak Collatz conjecture states that the iterated Collatz map (3x+1) has only one cycle on the positive integers.
Not independently verified, and probably easy to further optimise.  
Bigfoot  compiled  BB(7)  0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_1LB 
Bigfoot Comment  June 2024  @Iijil1 
Compilation of Bigfoot into 2 symbols, there was a previous compilation with 8 states 
Hydra  compiled  BB(9)  0RB0LD_1LC0LI_1LD1LB_0LE0RG_1RF0RH_1RA_0RD0LB_0RA_0RF1RZFile:Hydra 9 states.txt 
Discord message  June 2024  @Iijil1

Compilation of Hydra into 2 symbols, allconfirmed by Shawn Ligocki. @Iijil1 provided 24 TMs which all emulate the same behavior.
Previous compilation had 10 states, by Daniel Yuan, also confirmed by Shawn Ligocki. 
Beeping Busy Beaver
Cryptids were actually noticed in the Beeping Busy Beaver problem before they were in the classic Busy Beaver. See Mother of Giants describing a "family" of Turing machines which "probviously" quasihalt, but requires solving a Collatzlike problem in order to actually prove it. They are all TMs formed by filling in the missing transition in 1RB1LE_0LC0LB_0LD1LC_1RD1RA_0LA
with different values.