Usage of "many" vs "many a"?

Can someone please elucidate the difference between "many" and "many a"? In what context of usage should we add an extra "a" beside the word "many"?

For example:

• Many times, I had seen that . . . .
• Many a times, I had seen that . . . .

I do not know if the example is accurate enough to support my question, but I would like to know the difference between how they are used.

Solution 1:

Many a is a somewhat archaic or poetic or literary way of saying many.

Many times I had seen her in my dreams.

Note that it takes a singular complement:

Many a time I had seen her in my dreams.

EDIT

Just to throw another stick of wood onto this fire, there's another similar formulation frequently seen in literature:

Many's the time I've seen her in my dreams.

And I'll also add a quote from Shakespeare (The Merchant of Venice):

Shylock: Signior Antonio, many a time and oft / In the Rialto you have rated me / About my moneys and my usances

The question this raises is, did Shakespeare recognize a difference between "many a time" and "oft"? Did it mean something more to him than how we would see it today, as interchangeable with "often"? Or was he gratuitously padding out a line to fit the meter?

Solution 2:

Both many and many a convey the same meaning that is "a large number of". The only difference is that many is used with countable plural nouns followed by plural verb while many a is followed by a singular countable noun and takes a singular verb with it. E.g.:

1. Many soldiers were killed in the war.
2. Many a soldier was killed in the war.

Both statements have the same meaning. Many a is used in formal sense.