Open Source MicroApps
Tasty -- a full-featured tagging service
Fozzy -- full-text search service built on top of PostgreSQL's tsearch2 engine
Pubcookie -- single signon system.
W3C XSLT Servlet -- does XSLT transformations as a service
Solr -- Lucene based full-text search. (though they really should use HTTP DELETE to delete entries instead of POST)
GrassyKnoll -- Another Lucene based full-text search engine
YARS -- RDF/N3 data store in Java
Twister -- Random number generation for common statistical distributions.
CouchDb -- Distributed Document Oriented database with REST + JSON API (Probably the most advanced Microapp yet)
Drinkme -- Simple image resizing and thumbnailing.
Pita -- Data Pockets. Pretty much the prototypical microapp.
NQDQ -- a Queue/Stack microapp. Similar idea to amazon SQS, but not as heavy duty.
Gossip -- a microapp implementation of a discussion board or threaded comment system.
Gild -- a simple REST service which converts text in one of several simplified markup formats to html. Similar to rst2a, but supports markdown and textile formats and only outputs plain HTML.
Torsten -- an image tiling microapp
Commercial MicroApps
Programmable Web tracks web 2.0 APIs, and their corresponding mashups, including many apps which can be considered MicroApps.
Amazon's S3 -- online storage through a REST interface
Amazon's SQS -- a Queue service with a REST interface
Captchas.net -- anti-spam service
Free hosted microapps
Geonames -- nice REST service for looking up zip codes, lat/long, placenames, elevation, etc. JSON and XML output.
Triplr -- a service that generates RDF Triples from URIs
mailhook -- related at least. it's a service that listens for email and forwards it as a POST request to a web app that you set up.
rst2a -- reStructuredText to anything converter
See also Pending Microapps.