You can use « mince » and “zut” as these are not vulgar, yet do not connote innocence, at least for the time being for some people that still use those words.

“Zut” (as “mince”) has lost its power of expressiveness among men since roughly the 1960’s (ngram) but remains the most polite choice in writing and in speaking; you can use it in a public speech, whereas you will smear badly your image if you utter “merde” in its place. Surprisingly, although there has been a steep increase in the use of “merde”, lately the frequency of “zut” and “mince” have been going up. Notice that “crotte” and “flûte” as a curse words (very mild for the latter) have always been used very little.

There is an essential difference: you do not say “mince” in reply to someone so as to express to them a strong disapproval for what they say or think, you use “zut”; it’s the polite way to say “merde”; you might often be able to go on exchanging after saying “zut”, although still in an acrimonious manner, whereas the more vulgar word is much more likely to bring things to a head. You use “mince” in exclamations concerning facts themselves;

• Zut/Mince ! J’ai oublié mes clés dans la voiture.
• — Alors vous ne voulez pas me donner de chocolat au lait ?
  — Zut ! Vous le renversez sur la table et vous ne nettoyez rien ! Plus de chocolat au lait !

Examples from the TLFi

• Il a fait des pieds de nez et dit zut aux Allemands avec tant de crânerie (…) qu’on l’a élu à l’unanimité.
• J’ai dit zut à Fine qui ne voulait pas me donner une seconde tartine.

“Flûte”, in my opinion carries the stamp of innocence but as for the other terms it’s often a matter of the stamina and conviction that can be read in the action of the person using the term for it to confer force of expression; nevertheless it seems to be a word some girls and some women prefer, it is not so much used by men.

(TLFi, C) Interj., fam. Marque l’impatience, la déception. Synon. zut!

• Eh! flûte! s’exclama Bobouroche qui prit congé de la caissière en ces termes plutôt concis.
• Des flûtes. Synon. du flan!

“Crotte” might well be deemed to be a synonym of “merde” with an attenuated force, (“merde” is simply vulgar), but nevertheless it retains strongs connotations with excrement, and is in my opinion vulgar enough for one to want to avoid it;

(TLFi, A1) Interjection d’impatience, de dépit, de fureur. Crotte! (synon. atténué de « merde »).