All of these essentially mean the same thing - maybe, but here are some observations for each:

«Tal vez» is your standard way to say maybe, and it's used in all parts of the Spanish speaking world. The one-word version, «talvez», is not incorrect, but not as common as the two-word version. Most natives think the only correct form is «tal vez». Tal vez no sepan que me encantan los espárragos.

«Quizá(s)» is also used in all countries and just bit more formal than «tal vez». People tend to use «quizá» before a word starting with a consonant (quizá para ella significa/signifique mucho), and «quizás» with words starting with a vowel (quizás a mí no interesa/interese mucho), but this is certainly not a rule. Literature, especially older literature, tends to favor «quizá(s)», so it does have that "perhaps" flavor.

«Puede ser» and «puede que» are used in all countries and means possibly, especially when used as a stand-alone reply to a statement. —¿Te gustaría salir en un programa de televisión? —Puede ser. As for «puede que», you can use it in sentences and more colloquially as a stand alone response. —Puede que mi hermano no haya escuchado bien a la maestra y por eso no hizo la tarea. 

—¿Crees que Pedrito no haya entendido lo que dijo la maestra? –Puede que. Note that «puede que» calls for subjunctive.

«A lo mejor», to me at least, has more of a Mexican and Central American flavor, but I learned it's widely used elsewhere, including Spain. It means maybe. It does not mean "at best." For" at best," we say "a lo mucho" or "en el mejor de los casos." It doesn't have a positive connotation, either, despite the word "mejor" in the expression. It is a full synonym of maybe: A lo mejor mi abuelita se cayó porque el piso estaba muy resbaloso.

«De pronto» has a more Colombian and colloquial flavor, for sure. It is a full equivalent of «tal vez»: ¿Será que de pronto llueve hoy? Está muy nublado. It does not mean "suddenly" or "all of a sudden." For Colombians, that would be "de repente". For other countries, «de pronto» is a synonym of «de repente.» Colombian «de pronto» is, however, very much understood by many other countries.

«Igual», besides having the meaning for "same" or "I don't mind" as in "me da igual", can also mean maybe very colloquially. An example would be: ¿Por qué no se lo dices? Igual (tal vez) no lo saben todavía. In Mexico, people who use it actually say «igual y» as in "Igual y he fallado, pero ¿dime quién no se ha equivocado?"

«Chance y» is a pure mexicanismo.The word chance in Mexican Spanish is taken from English chance (opportunity), which was brought into English by the French, where it means luck (la chance). This usage has an informal and slangy flavor to it: Chance y pase matemáticas si estudio mucho (Maybe I'll pass math if I study a lot). When not used for maybe, it means oportunidad as in "Dame chance de ver si ya me cayó la quincena (en mi cuenta del banco)."

«Capaz (que)» is regionally used in the Andes and Rio de la Plata regions, especially Argentina, although I've heard it in some rural areas of Mexico and from some Venezuelans, too. It is considered informal. –Capaz (que) te salen con una historia en vez de contarte la verdad. 

«De repente» is not as well-contained geographically as «de pronto», but it's definitely used informally as a synonym of Colombian «de pronto». This is actually the reason why «de pronto» is so widely understood. While the Real Academia Española says it's used in Uruguay and Venezuela, it's usage extends beyond and can also be heard in these other countries: Argentina, Perú, México. Here's an example: De repente si llueve, se suspende el asado. In writing, it's generally avoided, since it also means “suddenly" depending on the context. Type in the comments if you also use it, and what country you're from, or where you learned Spanish.

If you're beginning to learn Spanish, you might want to stick to the first two, possibly three. If you're a native speaker or more advanced in Spanish, please comment on these forms or others you've heard! Also, don't forget to browse my other articles in the blog!