Q:

How difficult was it to apply for time-turners?

As far as we know from the books (I haven't been on Pottermore or seen any of JKR's interviews) Hermione is the only witch or wizard to have legally used a time-turner. 
So how does the application process work for being able to use one? And what is the criteria to allow them?

A:

There's remarkably little info about the process by which one can acquire a time-turner. Not only does the Ministry jealously guard them on their own premises but there's evidently a strict procedure needed in order to get hold of one.

The user must be of impeccable character
The user must not use it to aggrandise themselves
The user (if underage) must be vouched for by a senior wizard. In Hermione's case both McGonagall and Dumbledore are evidently involved in her getting the time-turner.
The time-turner is for use for a limited period only, to achieve a specified goal.
Abusing the time-turner is punishable (punishment not known).

‘It’s called a Time-Turner,’ Hermione whispered, ‘and I got it from
  Professor McGonagall on our first day back. I’ve been using it all
  year to get to all my lessons. Professor McGonagall made me swear I
  wouldn’t tell anyone. She had to write all sorts of letters to the
  Ministry of Magic so I could have one. She had to tell them that I was
  a model student, and that I’d never, ever use it for anything except
  my studies … I’ve been turning it back so I could do hours over again,
  that’s how I’ve been doing several lessons at once, see? But …
  Harry Potter and the Prisoner of Azkaban

Pottermore adds a little more info : 

The time-turner cannot be used for travel beyond 5 hours
The turner can only be used for "trivial" purposes due to the risk to the timeline.
There are strict laws and penalties for abusing the turner

The Ministry of Magic had ... the most stringent laws and penalties
  ... around those studying time travel.
Even the use of the very limited amount of Time-Turners at the
  Ministry’s disposal is hedged around with hundreds of laws. While not
  as potentially dangerous as skipping five centuries, the re-use of a
  single hour can still have dramatic consequences and the Ministry of
  Magic seeks the strictest guarantees if it permits the use of these
  rare and powerful objects. It would surprise most of the magical
  community to know that Time-Turners are generally only used to solve
  the most trivial problems of time-management and never for greater or
  more important purposes, because, as Saul Croaker tells us,
“ just as the human mind cannot comprehend time, so it cannot
  comprehend the damage that will ensue if we presume to tamper with its
  laws.

