Uncountable nouns

When is it possible to use an indefinite article before uncountable nouns? Only when they are defined in some way?

• music, art, love, happiness
• advice, information, news (It was a good advice/information/news)
• furniture, luggage (an old furniture, a heavy luggage)
• rice, sugar, butter, water (a/one rice on my plate)
• electricity, gas, power (a dangerous gas)
• money, currency

It depends on the noun. Some mass nouns are also count nouns. Take art, beauty and gas, for instance. You can say

The art of the native peoples was interesting.
Beauty is in the eye of the beholder.
Gas permeated the room.

or

He made lying into an art.
That cameo brooch was a real beauty.
The noble gases include helium, neon, argon, and xenon.

These differences rely on different senses of the word.

Some nouns use counters to turn them into count nouns.

three grains of rice (not "three rices")
a piece of toast

Some require units of measurement instead of counters.

200 kilowatts of electricity
a glass [or bucket, or liter, or cup, or cistern, etc.] of water

The point is, all this is part of learning vocabulary. There is no general rule covering all instances, and while you may find lists of count nouns and mass nouns, such lists will almost certainly be incomplete and even misleading in many cases.

The notion of "uncountable noun" will carry you only so far, as your question suggests. It is probably more useful to consider "countability" and "uncountability" as aspects under which any noun may be used.

For instance, from your list: music, love, sugar are all, ordinarily, "uncountable". Under the much/many test, we say:

I listened to much music
Babies need much love
She put much sugar in my coffee.

But they may also be employed under the "countable" aspect:

*I listened to many musics: folk music, classical music, pop music, &c. Is there a music which can be characterized as wholly atonal?
*Having suffered through many loves, he sought a love which should be pure, unsullied with ego.
*There are many sugars: fructose, glucose, sucrose, maltose, &c. Fructose is a sugar.

Any nominally "uncountable" noun may be employed as "countable" if it is regarded as a class with at least two members.

And with sufficient ingenuity (and at risk of some preciosity) it is possible to employ any nominally "countable" noun as uncountable by treating it as a quality rather than an individual:

There was much king latent in Hamlet; as Horatio observes, "He was likely [...] to have proved most royally".
Anything of wood still preserves much tree.
Suppressing the rebellion was a matter of much stick, little carrot.

EDIT: FumbleFingers objects (although he is too much of a gentleman to say so) that these last examples are factitious and implausible. It is quite true that I made them up as something one might say (but probably should not), and that they have a very 1920s "literary" quality about them.

But consider these, which appear to be perfectly spontaneous and colloquial:

from a YouTube comment (not suitable for work, or much else either): Hey now he thinks hes King....lol!...hes probably had too much king in him....hahahahaha

from Twitter: theres just so much tree in Cali man smh lol

from a football forum: why dosnt ** whinger get as much stick as benitez in the media

And if you object that stick in the last is just a slang term for criticism, consider this paper from the American Council for an Energy-Efficient Economy:

Too Much Stick? Not Enough Carrot? Testing the Presumption of Non-Compliance, by Eric Swan and Jarred Metoyer, RLW Analytics. Wim Bos, Sacramento Municipal Utility District

I love the English language.