First of all,diminished second would sound just like root
so you can have that only if composer is trying to mess with our brains.
Per example,in key of A you get note Bbb,that would be dim 2nd.
As for 3rd you have similar idea,dim 3rd would be just like 2nd
so once again it's not that often in use except if composer
really wants to write it that way.
Augmented is usually related to 5th actually.
Aug chord contains root,major 3rd and sharp(aug) 3rd.
This chord is used a lot in Jazz on 5th degree,like a dominant chord.
In key of D or Dm per example,you have A aug chord on 5th degree,
notes are A,C# and E#.
Now,this E# might confuse you cause it's the same thing as F note,right?
Here's the deal.
All of these chords have ordinary structure based on root,3rd,5th etc.
Now,starting from A, 5th has to start with letter E,
so you can have Eb,E ,E# etc,depends of type of 5th.
And if you say F then you have another aug chord but not A,it's F aug chord this time!!
It's in inversion cause its original shape is F,A and C#.
Now you are realizing one other thing,similar like dim7 chord.
Aug chord is actually made of 2 major 3rds intervals,
if you continue higher you'll get root note once more.
In dim 7 chord you have structure of 3 minor 3rds and next
note higher is root again.
Hope this helps somehow.