Ridding Twitter of Annoyances
Fact: Life is tough when you're easily irritated online.
Twitter is an amazing service (if used properly). It's a great way of getting news quickly, and personally, I find it to be a great place to find entertaining people. It definitely has its uses.
With that said, I have to say that every one of us have been annoying to someone, somewhere, at some time. It happens. However, with certain folks, it happens more regularly than others. Look - that's fine. That's what makes us all different. It's also what makes me find you annoying.
The most common solution is unfollowing someone (or taking it a step further by blocking and reporting them for spam). Easier said than done. Sometimes you have a couple conversations with a person and fool yourself into thinking you're friends. It happens. So what to do?
TweetDeck has recently (version 0.34+) implemented an amazing feature: Global filter

A Couple Essentials
Bieber

Let's face it. Justin Bieber exists. He also commands an legion of hormone-crazed teenagers online. Add his name to that filter list, as shown above, and you should be alright though. It's actually a fortunate thing that he has a unique name. So you should be safe to filter it as a word completely.
Formspring

Formspring is a service that allows people to ask questions anonymously. Seems very popular. I've noticed that this happens among people who seem to think they're more famous than they actually are. I don't have any intention of asking you anything.
The easiest solution is to add "www.formspring" as a keyword.
Knitting

No real explanation needed for this one. Just block that keyword and be done with it.
Exaggeration and Excess-Enthusiasm

It's a tough one figuring out the ideal number of exclamation-marks. Which number is ideal? I'm not entirely sure, but 6 has worked well for me (!!!!!!)

Again, it's tough to say how many is the ideal number to filter here. My studies have shown that 5-S's (sssss) is too much.

This is usually an indication that you should probably be following more mature Tweeters. Otherwise, "meeee" is a great keyword to filter.
Affection

"<3": You're better off unfollowing this person completely.

"♥": I wish I could block this, but TweetDeck seems to have some sort of unicode limitations - you might have to put up with this kind of Tweet for the time being.
People using Twitter as an IM service
You've seen this happen. Mutual-friends reply back and forth, then keep going for about 30 or 40 tweets. Amazing. I hate you. Solution: Add both their names to the keyword filter-list with an '@' as a prefix. For example: @AnnoyingPerson
Sources worth filtering
Foursquare

Foursquare is a service that allows people to act exactly like dogs pissing on trees marking their territory. Everybody fights for the "Mayor" position at a restaurant by "checking-in" regularly.
Block it by adding "foursquare" to your 'sources' filter-list.
Twitterfeed
Some Twitter accounts are linked to blogs. Each time a new article is published, a Tweet is published at the same time. I have an RSS reader, and I'm not scared to use it.
API

API almost always is a post by a bot.
Remaining Challenges
"your" versus "you're"
Also known as the Inability to understand the difference between "your" and "you're". If done consistently, this serves as a sure-fire way of determining a person's intellectual capacity. Unfortunately, I haven't been able to find a way of filtering this kind of stupidity other than unfollowing the person completely.
High-frequency Tweeters
Again, I have no solution for this. You might just have to add them to the from-people-list.
Conclusion
This is by no means an exhaustive list. Your annoyance-threshold may vary. If you have more suggestions, please leave a comment below.
I hate Internet Explorer
Any and all versions for that matter.
Text rendering in Firefox:

Text rendering in Google Chrome:

Text rendering in Internet Explorer 8 - WTF is this?!

Ok, I admit: I'm using the text-shadow property... but still!
Flickr badges, awards, invites and other nasty stuff
If you use Flickr, you've probably encountered images, flashing GIFs and badges in people's comments:


Cheap. Nasty. Whatever you'd like to call it, they're an eye-sore. I delete them whenever I see them on my photos.
Then I ran into this:

I have to admit, that's the funniest Flickr group I've ever seen - with a badge to match.
You can view the group here: Giant T-Rex Group
How to get rid of your CAPTCHA and still avoid spam
CAPTCHAs. You hate them, I hate them. Everyone's seen them and everyone will tell you they're annoying.

I'll admit, I used to use a Maths CAPTCHA, cause I thought it'd be easier for someone to add 5 + 6 than to read some scew text-characters. Apparently this only annoyed people more though (there is an amazing aversion to Mathematics out there). I can't blame you, it really is annoying.

And this one is very popular. Which apparently uses CAPTCHAs to help digitize books. That's great, but I don't feel like helping.

Which all brings me to something I'd encourage more people to use.
Spambots fill up fields in a form and submit. If there's a CAPTCHA, advanced spambots actually solve it and enter a value. Which leads to more advanced (nastier-looking) CAPTCHAs.
Invisible Defender is an example of spam-protection that does things a little differently. Instead of displaying a CAPTCHA, it adds a bunch of generic fields to the form. These fields remain invisible/hidden from the typical user:

(Most) Spambots don't realise this and fill up those fields too. The script detects if the fields were filled in and if they are, it shows an 403 error-page to the bots. I heard of this simple-concept about a year ago and thought it was such an eloquent solution. It almost seemed too simple to work.
Mind you, spambots are getting smarter, so some get past this too. But so far it's worked pretty well for me. I'm curious to know about your experiences too!
Facebook lacks a simple feature
How do you search for all of your friends that are from or in a specific location (e.g. all my friends in Namibia right now)?
Sounds like a simple enough search.
Or is this feature already implemented?
Update:
It's not linked to from the front-page - only when you do a search: http://www.facebook.com/advanced.php
Update:
And as a commenter pointed out, http://www.facebook.com/advanced.php doesn't exist anymore!
Gmail invites

This used to make sense when you couldn't get a Gmail account unless you got an invitation from someone who already had an account.
Now you can sign-up from the frontpage:

Now I have a (minimised) invitation box cluttering up my Gmail. I wish there was a setting to disable it (and I don't mean through a Greasemonkey script).
Feedburner counters
This (from 2008):

Reminds me of (from 1993):
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Definitely Web 2.0's version of the page-hit counter; definitely an easy way to make any professional-looking website look amateur.