A hiccup is two events, and the famous cures treat neither

A hiccup has two parts, and separating them explains the thing entirely:

1. An involuntary movement of the diaphragm, the muscle at the base of the
   lungs that does most of the work of breathing.
2. A quick closing of the vocal cords — which is what makes the "hic".

So the noise is not the hiccup; it is the second half of one.

What sets them off

Hiccups can start and stop for no obvious reason. Often something is
irritating the diaphragm:

• Eating too quickly, or too much
• Hot or spicy food
• Alcohol
• Carbonated drinks
• A bloated stomach
• Feeling nervous or excited
• Certain medicines
• Abdominal surgery
• Diseases that irritate the nerves controlling the diaphragm
• Metabolic disorders, and central nervous system disorders

The first six are why hiccups cluster around meals and parties.

The cures, honestly described

Hiccups usually stop on their own after a few minutes. You have heard the
remedies:

• Breathing into a paper bag
• Drinking or sipping a glass of cold water
• Holding your breath
• Gargling with ice water

There is no proof that any of them work. They are also not harmful, so
there is no reason not to try them — which is a more useful thing to know than
either "they work" or "they are nonsense".

When it stops being trivial

Some people have chronic hiccups: hiccups lasting more than a few days, or
returning repeatedly. These interfere with sleeping, eating, drinking and
talking, and at that point they are worth a provider's attention.

If a condition is causing them, treating that condition may resolve them.
Otherwise the options are medicines, surgery and other procedures — a
genuinely different register from a glass of cold water, and the reason the
duration is the thing to watch.

Source: MedlinePlus, U.S. National Library of Medicine.