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This is a community wiki of the answers in the comments.

• The compact Hausdorff topologies do not generally form a maximal antichain. If X is infinite, split X into two infinite halves and put the discrete topology on one half and the indiscrete topology on the other half. (Comment by François G. Dorais)

• There is a maximal compact topology on a countable space which is not Hausdorff. See Steen & Seebach 99. (Comment by Gerald Edgar)

• There is a minimal Hausdorff topology on a countable space which is not compact. See Steen & Seebach 100. (Comment by François G. Dorais)

• Those examples can be lifted to any cardinality space, simply by using the disjoint sum with any given compact Hausdorff space. (Comment by Gerald Edgar)

• Every set admits a compact Hausdorff topology, by topologizing it as the one-point compactification of the discrete space structure on the complement of any point. (Answer below by Cameron Buie)

(Feel free to edit and expand)

This is a community wiki of the answers in the comments.

• The compact Hausdorff topologies do not generally form a maximal antichain. If X is infinite, split X into two infinite halves and put the discrete topology on one half and the indiscrete topology on the other half. (Comment by François G. Dorais) Addendum: Without sufficient Choice, the infinite set $X$ may be amorphous. Amorphous sets are precisely the infinite sets for which this approach doesn't work. Very little Choice is needed to ensure that no such beast exists. (Edit by Cameron Buie)

• There is a maximal compact topology on a countable space which is not Hausdorff. See Steen & Seebach 99. (Comment by Gerald Edgar)

• There is a minimal Hausdorff topology on a countable space which is not compact. See Steen & Seebach 100. (Comment by François G. Dorais)

• Those examples can be lifted to any cardinality space, simply by using the disjoint sum with any given compact Hausdorff space. (Comment by Gerald Edgar)

• Every set admits a compact Hausdorff topology, by topologizing it as the one-point compactification of the discrete space structure on the complement of any point. (Answer below by Cameron Buie)

(Feel free to edit and expand)

This is a community wiki of the answers in the comments.

• The compact Hausdorff topologies do not generally form a maximal antichain. If X is infinite, split X into two infinite halves and put the discrete topology on one half and the indiscrete topology on the other half. (Comment by François G. Dorais)

• There is a maximal compact topology on a countable space which is not Hausdorff. See Steen & Seebach 99. (Comment by Gerald Edgar)

• There is a minimal Hausdorff topology on a countable space which is not compact. See Steen & Seebach 100. (Comment by François G. Dorais)

• Those examples can be lifted to any cardinality space, simply by using the disjoint sum with any given compact Hausdorff space. (Comment by Gerald Edgar)

• The Axiom of Choice implies that every set admits a compact Hausdorff topology, using the order topology of a successor well-ordering of it. (Comments by François G. Dorais and Gerald Edgar)

(Feel free to edit and expand)

This is a community wiki of the answers in the comments.

• The compact Hausdorff topologies do not generally form a maximal antichain. If X is infinite, split X into two infinite halves and put the discrete topology on one half and the indiscrete topology on the other half. (Comment by François G. Dorais)

• There is a maximal compact topology on a countable space which is not Hausdorff. See Steen & Seebach 99. (Comment by Gerald Edgar)

• There is a minimal Hausdorff topology on a countable space which is not compact. See Steen & Seebach 100. (Comment by François G. Dorais)

• Those examples can be lifted to any cardinality space, simply by using the disjoint sum with any given compact Hausdorff space. (Comment by Gerald Edgar)

• Every set admits a compact Hausdorff topology, by topologizing it as the one-point compactification of the discrete space structure on the complement of any point. (Answer below by Cameron Buie)

(Feel free to edit and expand)

This is a community wiki of the answers in the comments.

• The compact Hausdorff topologies do not generally form a maximal antichain. If X is infinite, split X into two infinite halves and put the discrete topology on one half and the indiscrete topology on the other half. (Comment by François G. Dorais)

• There is a maximal compact topology on a countable space which is not Hausdorff. See Steen & Seebach 99. (Comment by Gerald Edgar)

• There is a minimal Hausdorff topology on a countable space which is not compact. See Steen & Seebach 100. (Comment by François G. Dorais)

• The Axiom of Choice implies that every set admits a compact Hausdorff topology, using the order topology of a successor well-ordering of it. (Comments by François G. Dorais and Gerald Edgar)

(Feel free to edit and expand)

This is a community wiki of the answers in the comments.

• The compact Hausdorff topologies do not generally form a maximal antichain. If X is infinite, split X into two infinite halves and put the discrete topology on one half and the indiscrete topology on the other half. (Comment by François G. Dorais)

• There is a maximal compact topology on a countable space which is not Hausdorff. See Steen & Seebach 99. (Comment by Gerald Edgar)

• There is a minimal Hausdorff topology on a countable space which is not compact. See Steen & Seebach 100. (Comment by François G. Dorais)

• Those examples can be lifted to any cardinality space, simply by using the disjoint sum with any given compact Hausdorff space. (Comment by Gerald Edgar)

• The Axiom of Choice implies that every set admits a compact Hausdorff topology, using the order topology of a successor well-ordering of it. (Comments by François G. Dorais and Gerald Edgar)

(Feel free to edit and expand)