You can access web services in MiniScript using the http class. Let's gather here references to services that might be useful (and ideally, are free) for your Mini Micro games. I'll start:
Pastebin is normally used for sharing snippets of code or data in JSON format. But they have a very simple and generous web API, which lets you automatically post data to their servers, list all the items that have been posted for a particular user (where "user" to us really means channel — you can freely create these as needed), and of course get the data of any post. You could use this to store high scores, turn data in a turn-based game, player accounts and stats, etc.
Do not abuse this service; it's meant for relatively small amounts of data (a kb or two at most), with posts and fetches no more than once every few seconds per player. But for cases where that fits your game, this is a pretty great solution — simple, ready to use, and completely free.
If you find any other web services that might be useful in a Mini Micro game, post them below!