user14743: According to the ten fetters model, one of the five fetters which are only removed at attaining arahantship is ignorance (avijja). This means that anagamis are still subject to ignorance in some way. But it's also evident that anagamis possess a lot less ignorance than ordinary people. For example, belief in a self (sakkaya-ditthi) is abandoned at the first stage of enlightenment. So which forms of ignorance would an anagami still experience, and which forms not? Or is ignorance something that only sometimes arises in an anagami, the same way ill-will is only sometimes present in ordinary people?
Thanks in advance for your answers!

user13709: The one sutta that came to mind regarding your request had neither avijja nor anagami, but I think it may have an answer to your question according to my reading of the sutta:
In S​N​22.89, we read about Venerable Khemaka, who is ill and dealing with the conceit, "I am".

For when it comes to the five grasping aggregates I’m not rid of the conceit ‘I am’. But I don’t regard anything as ‘I am this’.

Because of this, we might infer that Ven. Khemaka was a non-returner. Yet Khemaka understood the escape even though he was ignorant of its fruition:

Although a noble disciple has given up the five lower fetters, they still have a lingering residue of the conceit ‘I am’, the desire ‘I am’, and the underlying tendency ‘I am’ which has not been eradicated.

The subtlety of ignorance in non-returners is compared with the lingering of a cleansing scent:

Although that cloth is clean and bright, it still has a lingering scent of salt, lye, or cow dung that had not been eradicated.


user8157: Ignorance as the last fetter is merely a broad fetter including any type of attachment or self-view that still may arise. 
For example, the life of the anagami may be threatened with deadly violence and the idea "I am" may arise in the anagami just before the anagami is beheaded with a sword.
Or the anagami may see many individual life forms slaughtered in war and may think those life forms or aggregates are "beings" or "people". For example, an anagami may believe 3,000 "beings" ("satta") "died" ("marana") on 9/11 where as an Arahant would perceive 15,000 aggregates (khandha) or 18,000 elements (dhatu) ending on 9/11. 
There is a sutta (somewhere) where an anagami did not attain arahantship because they got excited about the Dhamma and had the urge to tell their "uncle" (other other relative). The anagami still viewed their "uncle" as "a being" or "person".  
Or an anagami may get attached to the Dhamma, such as imputing "self-view" or "personhood" upon an individual that misrepresents the Dhamma doctrinally.
Ignorance is not-knowing the 4 noble truths and not knowing the three characteristics. The 1st noble truth is summarised as "attachment to the five aggregates". The three characteristics are impermanence, unsatisfactoriness & not-self. Therefore, if an anagami attaches to anything as "self", ignorance remains. 

user471: According to MN 9 (below), as long as you still have fermentation or effluents, you would still have ignorance, and vice versa (also see this question). Ignorance is a very deeply ingrained and deeply lingering fetter.

From the origination of fermentation comes the origination of
  ignorance. From the cessation of fermentation comes the cessation of
  ignorance. .....
From the origination of ignorance comes the origination of
  fermentation. From the cessation of ignorance comes the cessation of
  fermentation.

There is a vicious cycle between ignorance and fermentation.
There seems to be four types of fermentations in the Dhammawiki page of fermentation:


The mental fermentation of sense-desire (kāmāsava), Ex: 'All is pleasant'
The mental fermentation of desiring existence (bhavāsava), Ex: 'Being is good'
The mental fermentation of wrong views (ditthāsava), Ex: 'My opinion is best'
The mental fermentation of ignorance (avijjāsava). Ex: 'Suffering exists not'


According to this wikipedia page on ten fetters, the Anagami still has the fetters of lust for material existence, lust for immaterial existence, conceit, restlessness and ignorance.
So, the Anagami would have overcome the mental fermentation of sense-desire (kāmāsava) and the mental fermentation of wrong views (ditthāsava). However, he would still be challenged by the mental fermentation of desiring existence (bhavāsava). I take it that conceit is related to bhavāsava.
What does this mean?
The Anagami has overcome self-view, attachment to rituals, doubt, sensual desire and ill will. But the Anagami still has the mental fermentation of wanting to exist. And this will cause ignorance to continue.
You know the Four Noble Truths very well, dispelled wrong views (including self-view), overcame sensual desire and overcame ill will. But when someone threatens your life, you may get scared. Why? It's because you still want to exist.

user-1: I've read that one time Ven Anuruddha being a non-returner talked to Sariputta and it went something like this:

"With clairvoyance that is purified and surpasses the human, I survey
the entire world-system. My energy is roused up and unflagging, my
mindfulness is established and lucid, my body is tranquil and
undisturbed, and my mind is immersed in samādhi. But my mind is not
freed from the defilements by not grasping.”
Venerable Sariputta then answers:
“Well, Reverend Anuruddha, when you say: ‘With clairvoyance that is
purified and surpasses the human, I survey the entire world-system,’
that’s your conceit. And when you say: ‘My energy is roused up and
unflagging, my mindfulness is established and lucid, my body is
tranquil and undisturbed, and my mind is immersed in samādhi,’ that’s
your restlessness. And when you say: ‘But my mind is not freed from
the defilements by not grasping,’ that’s your remorse. It would be
good to give up these three things. Instead of focusing on them, apply
your mind to the deathless.” - AN3.130 https://suttacentral.net/an3.130/pli/ms


14743: <p>According to the ten fetters model, one of the five fetters which are only removed at attaining arahantship is ignorance (avijja). This means that anagamis are still subject to ignorance in some way. But it's also evident that anagamis possess a lot less ignorance than ordinary people. For example, belief in a self (sakkaya-ditthi) is abandoned at the first stage of enlightenment. So which forms of ignorance would an anagami still experience, and which forms not? Or is ignorance something that only sometimes arises in an anagami, the same way ill-will is only sometimes present in ordinary people?</p>

<p>Thanks in advance for your answers!</p>
 13709: <p>The one sutta that came to mind regarding your request had neither <em>avijja</em> nor <em>anagami</em>, but I think it may have an answer to your question according to my reading of the sutta:</p>

<p>In <a href="https://suttacentral.net/sn22.89/en/sujato" rel="nofollow noreferrer">S​N​22.89</a>, we read about Venerable Khemaka, who is ill and dealing with the conceit, "I am".</p>

<blockquote>
  <p>For when it comes to the five grasping aggregates I’m not rid of the conceit ‘I am’. But I don’t regard anything as ‘I am this’.</p>
</blockquote>

<p>Because of this, we might infer that Ven. Khemaka was a non-returner. Yet Khemaka understood the escape even though he was ignorant of its fruition:</p>

<blockquote>
  <p>Although a noble disciple has given up the five lower fetters, they still have a lingering residue of the conceit ‘I am’, the desire ‘I am’, and the underlying tendency ‘I am’ which has not been eradicated.</p>
</blockquote>

<p>The subtlety of ignorance in non-returners is compared with the lingering of a cleansing scent:</p>

<blockquote>
  <p>Although that cloth is clean and bright, it still has a lingering scent of salt, lye, or cow dung that had not been eradicated.</p>
</blockquote>
 8157: <p>Ignorance as the last fetter is merely a broad fetter including any type of attachment or self-view that still may arise. </p>

<p>For example, the life of the anagami may be threatened with deadly violence and the idea "I am" may arise in the anagami just before the anagami is beheaded with a sword.</p>

<p>Or the anagami may see many individual life forms slaughtered in war and may think those life forms or aggregates are "beings" or "people". For example, an anagami may believe 3,000 "beings" ("satta") "died" ("marana") on 9/11 where as an Arahant would perceive 15,000 aggregates (khandha) or 18,000 elements (dhatu) ending on 9/11. </p>

<p>There is a sutta (somewhere) where an anagami did not attain arahantship because they got excited about the Dhamma and had the urge to tell their "uncle" (other other relative). The anagami still viewed their "uncle" as "a being" or "person".  </p>

<p>Or an anagami may get attached to the Dhamma, such as imputing "self-view" or "personhood" upon an individual that misrepresents the Dhamma doctrinally.</p>

<p>Ignorance is not-knowing the 4 noble truths and not knowing the three characteristics. The 1st noble truth is summarised as "attachment to the five aggregates". The three characteristics are impermanence, unsatisfactoriness &amp; not-self. Therefore, if an anagami attaches to anything as "self", ignorance remains. </p>
 471: <p>According to <a href="https://www.accesstoinsight.org/tipitaka/mn/mn.009.than.html" rel="nofollow noreferrer">MN 9</a> (below), as long as you still have fermentation or <a href="https://buddhism.stackexchange.com/questions/9414/what-is-effluent">effluents</a>, you would still have ignorance, and vice versa (also see <a href="https://buddhism.stackexchange.com/q/25874/471">this question</a>). Ignorance is a very deeply ingrained and deeply lingering fetter.</p>

<blockquote>
  <p>From the origination of fermentation comes the origination of
  ignorance. From the cessation of fermentation comes the cessation of
  ignorance. .....</p>
  
  <p>From the origination of ignorance comes the origination of
  fermentation. From the cessation of ignorance comes the cessation of
  fermentation.</p>
</blockquote>

<p>There is a vicious cycle between ignorance and fermentation.</p>

<p>There seems to be four types of fermentations in the Dhammawiki page of <a href="https://dhammawiki.com/index.php?title=Asava" rel="nofollow noreferrer">fermentation</a>:</p>

<blockquote>
  <ol>
  <li>The mental fermentation of sense-desire (<em>kāmāsava</em>), Ex: 'All is pleasant'</li>
  <li>The mental fermentation of desiring existence (<em>bhavāsava</em>), Ex: 'Being is good'</li>
  <li>The mental fermentation of wrong views (<em>ditthāsava</em>), Ex: 'My opinion is best'</li>
  <li>The mental fermentation of ignorance (<em>avijjāsava</em>). Ex: 'Suffering exists not'</li>
  </ol>
</blockquote>

<p>According to this wikipedia page on <a href="https://en.wikipedia.org/wiki/Fetter_(Buddhism)#Sutta_Pitaka&#39;s_list_of_ten_fetters" rel="nofollow noreferrer">ten fetters</a>, the Anagami still has the fetters of lust for material existence, lust for immaterial existence, conceit, restlessness and ignorance.</p>

<p>So, the Anagami would have overcome the mental fermentation of sense-desire (<em>kāmāsava</em>) and the mental fermentation of wrong views (<em>ditthāsava</em>). However, he would still be challenged by the mental fermentation of desiring existence (<em>bhavāsava</em>). I take it that conceit is related to <em>bhavāsava</em>.</p>

<p>What does this mean?</p>

<p>The Anagami has overcome self-view, attachment to rituals, doubt, sensual desire and ill will. But the Anagami still has the mental fermentation of wanting to exist. And this will cause ignorance to continue.</p>

<p>You know the Four Noble Truths very well, dispelled wrong views (including self-view), overcame sensual desire and overcame ill will. But when someone threatens your life, you may get scared. Why? It's because you still want to exist.</p>
 -1: <p>I've read that one time Ven Anuruddha being a non-returner talked to Sariputta and it went something like this:</p>
<blockquote>
<p>&quot;With clairvoyance that is purified and surpasses the human, I survey
the entire world-system. My energy is roused up and unflagging, my
mindfulness is established and lucid, my body is tranquil and
undisturbed, and my mind is immersed in samādhi. But my mind is not
freed from the defilements by not grasping.”</p>
<p>Venerable Sariputta then answers:</p>
<p>“Well, Reverend Anuruddha, when you say: ‘With clairvoyance that is
purified and surpasses the human, I survey the entire world-system,’
that’s your conceit. And when you say: ‘My energy is roused up and
unflagging, my mindfulness is established and lucid, my body is
tranquil and undisturbed, and my mind is immersed in samādhi,’ that’s
your restlessness. And when you say: ‘But my mind is not freed from
the defilements by not grasping,’ that’s your remorse. It would be
good to give up these three things. Instead of focusing on them, apply
your mind to the deathless.” - AN3.130 <a href="https://suttacentral.net/an3.130/pli/ms" rel="nofollow noreferrer">https://suttacentral.net/an3.130/pli/ms</a></p>
</blockquote>
