The most commonly used is probably “De rien”.

Slightly more formal is this one : “Je t’en prie / Je vous en prie”

Your “Pas de problème” is used also, and is more casual.

The usual answer to “merci” in French is “de rien” which has about the same meaning as “no problem” and translates to “it’s nothing”.

There are a few main ways to say “you’re welcome” in French:

• Je vous en prie / Je t’en prie

I feel this is a little more formal than the others, but is the canonical French response to “thank you”. In some sense, it can be equivalent to “don’t worry about it”.

• Pas de problème

Google translate was right. It is widely used, it’s informal and it likely comes from English in the first place.

• De rien

This phrase likely comes from the Spanish de nada. Rien means “nothing”, so it could be somewhat analogous to “don’t give it a second thought”. But shorter and more informal.

• Bienvenue

Used in Canada, this is the literal translation of “welcome”. In other French-speaking countries, it would likely not be understood. This use of bienvenue is an anglicism; its correct use is mostly one of greeting or of appreciation, like when you say, for example, that relief is welcomed.

Other ways to answer a “Merci” are:

• “Il n’y a pas de quoi“, sometimes abbreviated in “Pas de quoi“
• around Toulouse: “Avec plaisir“
• in Belgium: “S’il vous plaît“

À l’oral, après merci, en France (St Etienne) j’entends souvent répondre Y’a pas d’soucis.

In Northern France, I most frequently hear “Il n’y a pas de quoi”, or just “pas de quoi” (more casual).

I do use “avec plaisir”. I find those one (below) strange even if they are more common.
(What follows is just my very personal interpretation).

1. “Je t’en prie / Je vous en prie” sound a bit arrogant/posh,
   like: nooo please, it’s ok, don’t thank me dear, really, no I insist don’t, oh dear…

2. “Il n’y a pas de quoi” looks absurd, as if we needed a scale to
   thanks somebody. “No it’s not enough, you can’t thank me. (It sounds
   also like #1)

3. “Pas de problème” or “y’a pas de soucis” is the most common, it
   sounds like: you did not bother me too much, just a little, but it’s
   ok (but you did…)

4. “De rien”: do fit for guys who aren’t able to accept things (good or bad)
   from the others. “You don’t own me anything and I don’t own you
   anything, let’s stop talking”. (It sounds also like #3)

5. “Bienvenue” sound from quebec (so, funny), but it provides a warm answer, I like it.

Note that french are still very attach to their old formal way of talking: sometime spontaneity and warmth do make the french uncomfortable. But “avec plaisir” works with anybody, even if it sounds spontaneous and shows warmth it also somehow shows some formal “politeness”.

From simple : “De rien”, to more formal: “Pas de quoi” ou “il n’y a pas de quoi”, and more polite: “Je vous en prie”, “je t’en prie”. “Pas de problème” est plus récent et semble importé du sabir globish…