Am I allowed to bring mini knife in public in Singapore without applying?

I'm not sure how big your knife is from the picture. However, there is a list of items which are permissible or forbidden to import or possess.

Most categories of knives (hunting knives, diving knives etc.) are okay to have, however, you might not be able to carry them in public without a reason:

However unlawful possession of such permissible items in any public place may constitute to an offence of Possession of Offensive Weapon or Scheduled Weapon under the Corrosive and Explosives Substances and Offensive Weapons Act

So taking your diving knife on your way to the beach is probably okay, but you shouldn't take it to the mall.

Some other categories of knives are illegal to have or import (see the same list as above), this notably includes flick knives and gravity knives which look a bit similar to yours. However, those are capable of being opened by a button or by gravity and yours doesn't look like it can do that. (as far as I can tell from the picture).

Note that nowhere does it mention pocket knives (like Swiss Army knives), so those are not regulated. If your knife is one of those, you should have no problem importing it.

It is not very common to carry knives in public though, so if you get searched (which is very unlikely), you might have to explain why you are carrying a knife. So personally I would go along with what has been said in the comments to your question and recommend you to leave your knife in your luggage/hotel unless you plan to go hiking or something along those lines.