translate those word into English, 
竟然:means to one's surprise

ex: It's "extraordinary" that he managed to sleep through the party.
     真想不到他竟然从聚会开始一直睡到结束。

ex: It is "inconceivable" that the minister was not aware of the problem.

     令人难以置信的是那位大臣竟然没有意识到这个问题。

I think that's no corresponding word for 竟然 in English, 竟然 convey the expression of realizing something in a way that was not expected ( out of blue )

忽然 & 突然:suddenly、sudden

ex: She suddenly felt totally drained. 
     她突然(忽然)感到精疲力竭。 

ex:The sudden rise in share prices has confounded economists.
     股价的突然上涨使经济学家大惑不解。

basically, they (忽然 & 突然) are the same.

if a man come into a  Lady toilet.you can use竟然!

忽然and突然，this two words mean very be close to,and be similar...

my english is not good .haha

竟然 means surprise.   

The question was so simple, and yet he couldn't answer it.

这么简单的问题他竟然没回答上来。

sb.was not aware of the problem.

突然，忽然means suddenly

我突然想起来我上次是在什么地方见到他的。

It suddenly came back to me where I saw him last.

仍然，茫然，果然，虽然

Anytime : )