What evidence exists to support that Singapore is the world's
strictest nations? I know that Singapore is famous for levying
heavying fines for seemingly inconsequential offenses. Furthermore,
Singapore is rumored to have the world's "highest rate of capital
punishment per capita". (Is this true? How does it compare to other
countries?) Do we know why Singapore is such a strict country?

I just wanted to mention the one tourist item I gave every one of my
friends after my brief visit to the City/State was a "Singapore is a
FINE city!" that then listed a bunch of infractions-- chewing gum,
spitting, insulting the dignity of a lady, failing to sweep your
sidewalk, etc.

HI,
It appears that you may be right about Singapore's having the highest rate.

http://www.singapore-window.org/sw02/020412re.htm
http://www.singapore-window.org/sw03/030924a1.htm
http://web.amnesty.org/library/Index/ENGASA360012004?open&of=ENG-SGP

Why is Singapore such a strict country?
As you can read in Wikipedia, from its founding it has been pretty
much a one party country with for decades a president who was
successful in guiding the island country to become an independent
economic power, something that was not expected when it gained
independence.  This political situation allowed strict laws to
implemented.
Sure, they are drastic by western standards, but no one needs to spit
out chewing gum on the street, etc. and risk corporal punishment:
caning.  In the West, it is generally felt that the punishment should
be appropriate to the seriousness of the crime, rather than being
hefty as a deterent.  But it would seem that the deterent factor works
in Singapore, even if it goes against "our" way of thinking and
sometimes is applied to foreigners who did or did not know better. 
(Ignorance of the law not being an excuse is a principle that can be
traced back to biblical times.)
As to capital punishment:  it always existed there and in that part of
the world.  Again, it is not approved of in the West (see all the
headlines today about California, in European papers too), but once
again, except for crimes of passion, people do not have to commit
capital crimes, especially drug trafficking and possession, the major
crimes that result in executions.
Considering the fact that these do occur, I expect that the drug
problem in Singapore as a major international crossroads would be
much, much greater if the punishment were not draconian, and would be
a problem not just of the trafficking but also for the population.
Well, that is just one man's opinion, here is the Wikipedia site:
http://en.wikipedia.org/wiki/Singapore
Cheers, Myoarin