Features worth searching for:
Log-by-level. E.g. info, warn, verbose, error.
Log-by-module/theme. E.g. MyClass, file1.js, Data, UI, Validation, etc. Sometimes called “groups” or other things.
Log-errors. Info, warn, verbose are all the same data type, but the error is a complex data type and the work flow differs dramatically from the others.
Log-to-different places, e.g. HTML, alert, console, Ajax/Server
Log-formatting. E.g. xml, json, CSV, etc.
Log-correlation. E.g. if you log to 2 places, say a client, server and web-service and db, and a transactions passes through all four, can you correlate the log entries?
Log-analysis. E.g. if you generate *a lot* of log entries, something to search/summarize them would be nice.
Semantic-logging. E.g. logging (arbitrary) structured data as well as strings or a fixed set of fields.
Analytics. Page hit counters. (I didn’t search for these)
Feature Usage. Same, but for applications where feature != page
Console. Sometimes as a place to spew log entries, sometimes as a place for interactive execution of code.

Repository Queries:
http://bower.io/search/?q=logging 25 entries right now.

https://www.nuget.org/packages?q=javascript+logging Nuget JS logging libraries.

https://nodejsmodules.org/tags/logging – Nodejs Repository query

Various Uncategorized Browser-Centric Libraries
https://github.com/enoex/Bragi-Browser category logger (here they are called “groups”)
https://github.com/better-js-logging/console-logger category & level logger.
https://github.com/latentflip/bows colorful logging by category
https://oaxoa.github.io/Conzole/ Side console.
https://github.com/structured-log/structured-log – Serilog/Structured Log
http://log4javascript.org/ Log4JavaScript – for people who like the log4x API. As of 2015, appears dated & unmaintained.
http://www.softwarementors.com/log4js-ext/ More upto date log4x library. Fancy on screen log.
https://github.com/stritti/log4js A level-logger. (Many features are IE only)
http://smalljs.org/logging/debug/ Console logger with module filters.
http://jsnlog.com/ Client side logger that sends events to popular server side logging libraries (more than just server side node)
http://www.songho.ca/misc/logger/logger.html on screen (HTML) logging overlay
https://github.com/jbail/lumberjack Monkeypatches the built-in console object.

Microlibraries for Browser
These might not be any smaller than other libraries.
https://github.com/bfattori/LogJS – supports local storage logging.
https://github.com/kapilkaisare/sleeper-agent 4 level logging with an on/off switch at runtime.
https://github.com/mattkanwisher/driftwood.js – 4 level logging with environment switches & ajax
https://github.com/pimterry/loglevel. 4 level logging Supports plugins.
https://github.com/cowboy/javascript-debug/tree/v0.4 console.log wrapper
http://js.jsnlog.com/ Same as JSN Log, but just the JS part, so it’s like a microlibrary.
https://github.com/nhnb/console-log – Polymer/web-component style console logging
https://github.com/icodeforlove/Console.js – Polyfill for pretty-console display?
https://github.com/Couto/groundskeeper – Build step to remove console.log entries before sending to production.

https://github.com/pockata/blackbird-js -Abandoned? Not sure what it does.
NitobiBug -Abandoned. If you look long enough you can find websites that serve up the file.

Browser plug ins
http://getfirebug.com/logging – Firefox centric.

Error Logging
Error logging is more than a print statement. Generally, at point of error you want to capture all the information that the runtime provides.
http://www.stacktracejs.com/ Stacktrace and more.
http://openmymind.net/2012/4/4/You-Really-Should-Log-Client-Side-Error/ Roll your own

Node Loggers (might work in Browser, not sure)
https://github.com/enoex/Bragi-Node (Node Centric)
https://www.npmjs.com/package/winston (Node Centric)
https://github.com/jstrace/jstrace (Node centric)
https://nodejs.org/en/blog/module/service-logging-in-json-with-bunyan/ Bunyan

Commercial Loggers
Often a opensource client that talks to a commercial server. No idea if these can work w/o the server component.