$\text{ZFC}$ proves that each $\aleph_{n}$ for $n\in \omega$ is a regular cardinal. But it seems without the Axiom of Choice there are many consistent possible choices for cofinality of such cardinals.
Please introduce some references for any known consistency result with $\text{ZF}$ about possible cofinalities of $\{ \aleph_n\}_{n\in \omega}$.
Here are some partial answers:
1) By a result of Gitik, all $\aleph_n$'s can have cofinality $\omega.$
2) the paper "Cofinality and measurability of the first three uncountable cardinals" by Apter, Jackson and Löwe completely solves the problem for $\aleph_1, \aleph_2$ and $\aleph_3.$ The following is taken from their introduction:
"In this paper, we investigate all possible patterns of measurability and cofinality for [$\aleph_1, \aleph_2, \aleph_3$]. Combinatorially, there are exactly 60 such patterns of which 13 are impossible for trivial reasons (e.g., if $\aleph_1$ is singular, then $\aleph_2$ cannot have cofinality $\aleph_1$). In this paper, we prove that the remaining 47 patterns are all consistent relative to large cardinals.''
3) The following is appeared in Ioanna Dimitriou's PhD thesis:
Theorem. Assume V is a model of ZFC which contains $\omega-$many strongly compact cardinals. For any function $f: \omega \rightarrow 2$ there is a model of ZF in which $\aleph_{n+1}$ is regular if $f(n)=1$ and singular if $f(n)=0.$
Ioanna Dimitriou's Ph.D. thesis as well many papers by Arthur Apter would be a good start.
You may want to read the proof of the Feferman-Levy construction, there's a nice proof in Jech "The Axiom of Choice" as well Ioanna Dimitriou M.Sc. and Ph.D. theses. Also interesting is the results in
Truss, John "Models of set theory containing many perfect sets." Ann. Math. Logic 7 (1974), 197–219.
Also relevant are the results by Magidor that if $\operatorname{cf}(\omega_1)=\operatorname{cf}(\omega_2)=\omega$ then $0^\#$ exists (this can be found, I believe in Magidor's paper proving the covering lemma, but also in Jech "Set Theory" as an exercise for Chapter 18); and similar applications of the covering lemma which prove the necessity of large cardinals for some of the results. Related to that is Andres Caicedo's answer to my [very old] MSE question: Consistency of $\operatorname{cf}(\omega_1)=\operatorname{cf}(\omega_2)=\omega$.
On the same breath, one should mention
Busche, Daniel; Schindler, Ralf "The strength of choiceless patterns of singular and weakly compact cardinals." Ann. Pure Appl. Logic 159 (2009), no. 1-2, 198–248.
Which also deals with the question of many singular $\aleph_n$'s and its consistency strength.
From these papers and their reference, I think one should be able to cover all the known material.
