Is there an equivalent of Bottle or Sinatra for Perl6 or Perl5?

Dancer comes very close and aims to be Sinatra for Perl.

Mojolicious::Lite is another very simple approach on web development.

For Perl 5 there exists a lot of Frameworks:

• Catalyst
• Dancer
• Mojolicious
• Jifty

There are a lot more. What you also want to look at is at PSGI/Plack. PSGI is a general interface for communication between web servers and perl web frameworks. Plack is an implementation of it. One of the most used Framework at the moment is probably Catalyst. On the Plack page you found a lot more web frameworks.

Bailador and Cro are two quite advanced microservices frameworks in Perl 6.