Actually, it is pretty simple. If there is a vowel after “beau”, it becomes “bel”:

Il est bel et bien avocat.
Quel bel écureuil !
Quel beau cheval !

You can see more exemples and exercises here.

“beau” is used by default.

The form “bel” is used when the next word starts with a vowel.

For example:

• un beau rocher
• un bel arbre
• un bel et grand édifice

Yes, they are both the masculine form of beau, but bel is used when the noun after it starts with a vowel. For example,

Un bel appartement

Note that bel is before the noun. Follow the rule called BANGS word. B stands for Beauty, A stands for age, N stands for new/old, G stands for good, and S stands for size. If any adjective falls in a category of BANGS, then it will go before the noun. Usually they go after the noun.

There are very few adjectives with 2 forms: short, before a noun begining with a vowel, and another often bigger, to put after the noun.

And it is for masculine, singular.

I see 4 cases (with help of M. Glisse):

• beau / bel
• vieux / vieil
• nouveau / nouvel
• fou / fol

“Beau” and “bel” are both french adjectives of the some token. That is that ‘beau’ becomes ‘bel’ before a french singular masculine noun starting with a vowel or a mute ‘h’. Hence, if we take the nouns “Art” and “homme”, the plural of beau stays unchanged while becoming plural: “les ‘beaux’ arts” or “l’Ecole des ‘Beaux’ Arts de Paris”.

So, ‘beau’ keeps its form when qualifying a plural noun here, ‘arts’.

But if art was singular, it’s ‘bel’ that would has been used.

Example: “Quel ‘bel’ art !” or “Quel ‘bel’ homme !” and “Quels ‘beaux’ hommes !” for the plural.

While using the same process we would have said: “Quelle ‘belle’ femme !” for the singular, and “Quelles ‘belles’ femmes !” for the plural.

We will here use the same process for:

“Quel ‘vieil’ homme !” and “Quels ‘vieux’ hommes!”.

Where, ‘H’ is also mute.