How to find monologues and scenes:
1) Find plays you are interested in. Use the Find Scripts tab to learn how.
2) Look at the character discriptions to see if they have a character that fits you.
3) Find large chunks of text within the play where that character is speaking alone (for monologues) or with another character in an equal back and forth (scene). You can splice text together if you wish.
4) Examine that chunk of text for an arc that happens in the moment (they come to a realization, they change their minds, they resolve a conflict), rather than them recounting something that already happened.
Drama Online has some amazing tools to help you find monologues and scenes within their play collections. You can't find these plays on the open web, so these monologues will be more unique then those you find through a simple google search.
Here is a tutorial about how to use Drama Online's tools to find monologues and scenes:
There are also plenty of monologue databases available on the internet, though their ease of access may make the monologues within them likely to be overused.