What’s the right term for "expensive things that have no real use"

The rich people buy many luxury things like expensive cell phones, gadgets with brilliants, bags made of snake skin and other such things. They buy it in order to separate themselves from the “crowd” and to show others their social status and wealth. There are other people that want to look rich (or VIP) so they spend their lifetime scraping and saving in order to buy these expensive things. However, these things have no real use, these things are just expensive “trifles”. But many people are still sure that these things will make them happy!

The question is:
What’s the right American English term for “things that have no use which people buy, spending their lifetime, to show others their importance”.

Maybe trifles or trinkets are the right words?

So you are looking for the pricey version of a tchotchke. There are a couple in common use. Bling is an excellent fit to your criteria of expensive and highly visible. Trophy seems to be used especially when it refers to the toys (and girl/boytoys) of successful businesspeople.

I agree 100% with onomatomaniak. Avoid trinkets, baubles and trifles as they all imply that the items are cheap. I also agree with onomatomaniak's suggestion to use the term "luxuries". You could also go with the very similar term of "luxury items".

Another term which you might consider (and which I quite like myself) is "extravagances". Oxford Dictionaries Online defines the terms as "a thing on which too much money has been spent or which has used up too many resources". An nice example sentence (from Urban Dictionary) would be "Nouveau riche - low class wannabes showing off their wealth with enthusiastic extravagance (aka bling)." Note that you can also use this term as an adjective, e.g. in a phrase like "the extravagant lifestyle of the nouveau riche."

I think the term you may be looking for is positional good, which is the economist's term for things bought chiefly to prove you can afford them, rather than for any real use.

See also here.