According to Catholicism, Can God annihilate the soul? Catechism of the Catholic Church 
  
  362 The human person, created in the image of God, is a being at once corporeal and spiritual. The biblical account expresses this
  reality in symbolic language when it affirms that "then the LORD God
  formed man of dust from the ground, and breathed into his nostrils the
  breath of life; and man became a living being." Man, whole and
  entire, is therefore willed by God.
  
  366 The Church teaches that every spiritual soul is created immediately by
  God - it is not "produced" by the parents - and also that it is
  immortal: it does not perish when it separates from the body at death,
  and it will be reunited with the body at the final Resurrection.


The Catholic Church teaches that the soul is immortal because God breathed the breath of life into the man.

My argument here is that, if God created the man in His likeness and breathed His life into the man, it would mean that man or the soul was created to be immortal, just like God. If this is true, then it would mean that even God himself cannot reverse the immortality that was imparted into the man!

According to Catholicism, is God able to annihilate the soul? Can God make the soul vanish forever?
In the Summa Theologica (First Part, Question 104, Article 3), Aquinas asks &quot;Whether God can annihilate anything?&quot; He concludes,

Some have held that God, in giving existence to creatures, acted from natural necessity. Were this true, God could not annihilate anything, since His nature cannot change. But, as we have said above, such an opinion is entirely false, and absolutely contrary to the Catholic faith, which confesses that God created things of His own free-will, according to Ps. 134:61: &quot;Whatsoever the Lord pleased, He hath done.&quot; Therefore that God gives existence to a creature depends on His will; nor does He preserve things in existence otherwise than by continually pouring out existence into them, as we have said. Therefore, just as before things existed, God was free not to give them existence, and not to make them; so after they are made, He is free not to continue their existence; and thus they would cease to exist; and this would be to annihilate them.

In other words, Aquinas believes that since God creates things because he wants to, and because he loves them, we must believe that it was possible for him not to give things (including the soul) existence, and similarly at least theoretically possible for God to fail to continue them in existence.
However, the very next question he considers is &quot;Whether anything is annihilated?&quot; (emphasis added) His conclusion is:

Some of those things which God does in creatures occur in accordance with the natural course of things; others happen miraculously, and not in accordance with the natural order, as will be explained. Now whatever God wills to do according to the natural order of things may be observed from their nature; but those things which occur miraculously, are ordered for the manifestation of grace, according to the Apostle, &quot;To each one is given the manifestation of the Spirit, unto profit&quot; (1 Cor. 12:7); and subsequently he mentions, among others, the working of miracles.
Now the nature of creatures shows that none of them is annihilated. For, either they are immaterial, and therefore have no potentiality to non-existence; or they are material, and then they continue to exist, at least in matter, which is incorruptible, since it is the subject of generation and corruption.2 Moreover, the annihilation of things does not pertain to the manifestation of grace; since rather the power and goodness of God are manifested by the preservation of things in existence. Wherefore we must conclude by denying absolutely that anything at all will be annihilated.

In other words:

If God annihilates something, he will do it either (a) as part of the natural course of things, or else (b) as a miracle, to display his power or for some other teaching purpose.
But in nature, we see that no created entity is annihilated in the natural course of things. Either they are material, and the matter constituting them continues in existence even after they die or are worn down to nothing; or else they are immaterial, in which case (since they are pure form with no accident) it is impossible for them to fail to exist.
Furthermore the annihilation of entities does not constitute a display of God's grace and power; these are best displayed by God's maintaining things in existence:


For you love all things that are
and loathe nothing that you have made;
for you would not fashion what you hate.
How could a thing remain, unless you willed it;
or be preserved, had it not been called forth by you?
But you spare all things, because they are yours,
O Ruler and Lover of souls,
for your imperishable spirit is in all things!

(Wisdom 11:24–12:1)
Thus, since God does not annihilate anything either in the natural course of events or as a miracle, we must believe that nothing at all—including the soul—is ever in fact annihilated.

1 Psalm 134 in the Septuagint, 135 in the Hebrew numbering.
2 That is, Thomas claims, because beings are generated out of matter, and they degenerate and decay (they are corrupted) back into matter, the matter itself is not corruptible. Matter can be formed into new entities, and these entities can decay back into matter, but the matter itself is not created nor destroyed—it is not corruptible.