Unfortunately, there is no rule. Whether the h is considered a consonant (H aspiré, even though the letter itself is not sounded) or is transparent (H muet) is a case-by-case matter, partly driven by etymology. Even words of the same family are sometimes pronounced differently: l’héroïne [eʁɔin], l’héroïsme [eʁɔism] but le héros [‘eʁo].

It’s more common for words of Latin or Greek origin to have a H muet, and for words of Germanic or more recent origin to have a H aspiré, but this is by far not a hard-and-fast rule one way or another.

Wikipedia has a list of words with H aspiré, and that’s as good as it gets.

There are also a few words where the h is expirated (H expiré): interjections such as ha, hé, etc. and onomatopoeia (e.g. hou [hu] or [u], the cry of the owl).

Je pensais à verser ceci de la part de http://forum.wordreference.com/showthread.php?t=109445.

De l’utilisateur Gil : Moyennant http://www.aidenet.com/scolaire/grammaire01.htm sur « h » :

Consonne muette quand on ne l’entend pas dans la prononciation du mot : l’homme.
Consonne aspirée lorsqu’elle fait prononcer la voyelle suivante avec une légère aspiration : la haine.

De l’utilisateur pktopp :
“He’ros must be pronounced with h aspire. Otherwise, its plural form “les he’ros” would be identical in pronunciation to “les z’eros”.

Just found this quote in the DICTIONNAIRE DES DIFFICULTES DE LA LANGUE FRANCAISE (Larousse):

“héros se prononce avec h aspiré (pour éviter la liaison les-z-[h]éros [zéros]). Dans tous les dérivés (héroïne, héroïque, héroisme etc.) l’h est muet.”

This book is a must have for any scholar of French. There is a large section dedicated to h aspiré . Just look under “H”.

De l’utilisateur geve :

Found on http://membres.lycos.fr/clo7/grammaire/consonH.htm :
“Si héros a l’h muet, son féminin héroïne ainsi que tous ses dérivés (héroïque, héroïquement, héroïsme…) ne l’ont pas.”

D’après http://www.sdv.fr/orthonet/pages/informations_p5.html, les mots d’origine germanique ont un H aspiré, tandis que les mots d’origine grecque ou latine ont un H muet.

De l’utilisateur jann :

Correct, liaison is forbidden when the following word begins with an aspirate H. The aspirate H itself is just as silent as the normal “unaspirated” H. The only thing that changes is whether or not you get to make élision & liaison.

H non-aspiré
le + honneur –> l’honneur (pronounced [lɔnœ:ʀ], with élision in the singular)
les + honneurs –> les honneurs (pronounced [lεzɔnœ:ʀ] or [le-] because you make the liaison in the plural)
There are many, many words like this: l’hôpital, l’homme, l’hiver, l’histoire, etc.

H aspiré
la + haine –> la haine (pronounced [laεn], no élision in the singular)
les + haines –> les haines (pronounced [lεeεn] or [le-], no liaison in the plural)
There are only a few dozen words like this : le haricot, le hasard, la honte, etc.

When you learn a new word that starts with an H, how do you know which kind of H it is? If you can look at the word and recognize that it’s based on a Latin root, this might be a good sign that the H will be non-aspirate. But what if you don’t know enough about Latin to be able to guess the root? In this case, you can look in a dictionary. If the word starts with an aspirate H, the pronunciation symbol for the word will start with an apostrophe, or you will see something like init. asp. written after the pronunciation.

In general, I think it’s safe to say that the H at the beginning of a French word is usually not aspirate. Other than words borrowed from English (le hockey, le hold-up), there are really only a few dozen French words that start with aspirate H. You’ll find a pretty good list in a white box at the bottom of http://www.ieeff.org/212phonology.htm, or see OlivierG’s post above.

No, the French aspirate “H” of héros does not resemble the English H of “his”. French does not have an H sound like English “his.”

As I said in my post above, the aspirate H itself is just as silent as the normal “unaspirated” H. So the reason you cannot hear a difference between the two in conversation is because there isn’t one! The only thing that changes is whether or not you get to make élision & liaison… and this, you will certainly hear.