There are plenty of ways of getting tickets (including free ones) to events that have sold out. Some of them rely on you being famous, beautiful, well-networked or more-than-averagely-confident (or rather, less easily embarrassed by asking for stuff) but here are some suggestions that depend less on your current personal attributes.
Consider the phrase "this event is now completely sold out" to be a fun challenge and adopt the attitude of my grandfather who, according to my grandmother, understood "No entry" signs to mean "come on in Andrew" ;) Be polite, but tenacious.
1. Ring the box office
Venues sell tickets online as do other third party ticket sellers. If either type of site is showing that the event is sold out it's worth ringing up and seeing if there are any that aren't on the system. I've lost count of the number of lovely conversations I've had with box office people telling me that there's actually a glitch and they do have a few tickets left. Third party sites with unsold tickets will usually return these to the main venue for resale so information online can be a bit variable and occasionally inaccurate.
If they've really, really sold out then ask about returns (i.e. purchased tickets that people can no longer use). Ask how these are resold (first come first served, out of a hat, returned to the online booking system for anyone who spots them, offered to the first person on the waiting list etc).
Ring back in a day or two and see if anything's changed.
2. Go by yourself or sit apart from friends
It's often a lot easier to get one seat than it is to get two next to each other. When a show is very busy you may actually find it difficult to get one seat on its own. If it's part of a pair (i.e. surrounded by booked seats) then the online system might not permit just one to be bought but if you ring the box office they can sort that out.
3. Look for "win tickets to ..." tweets and newspaper competitions
If you're really keen you can set up a Google alert which will email you if certain keywords are published on the web, you can also keep an eye on relevant hashtags and the venue's own social media sites as they'll likely retweet opportunities to win tickets. If the event has sponsors it's worth seeing if they offer some freebies themselves.
Just seen this thanks to returned tickets last min. Ab fab. Lovely setting too. Tom Hollander is just bloody brilliant! https://t.co/c7a4N0JBhM— Ally Sinyard (@allysinyard) September 30, 2016
4. Spare tickets
a) turn up on the day / ring the box office before the show
Sometimes someone in a group won't be able to leave work on time, or they're a bit ill and so people outside the venue might have a spare ticket that they're trying to resell. Make sure you check the date of the ticket before handing over money though. Some venues take a dim view of these kinds of ticket resales so I suppose you might find you're refused entry - generally this depends on the cost and level of security involved. Worth checking beforehand. Be careful about touts and inflated prices.
The box office may have returns, where someone realises they can't make the event and hands back their ticket for resale - make yourself known to the box office on arrival (hopefully you'll previously have rung them to let them know of your interest and to get yourself on any waiting list).
b) people offering tickets online
People can also post their spare tickets to things like Twickets (sold for face value) and they will also mention spares on Twitter etc. There are other organisations like Scarlet Mist and Stub Hub who match ticket sellers to buyers.
5. Have a blog and ask for a press pass
This one probably requires the most work as you'll have to already have a blog in place for this to be credible and it might be a bit cheeky if you don't then write about the event. The more "official" the event the less likely this'll work as they'll probably already have their own list of press people. But worth a shot.
6. The venue might release more tickets, put on more shows
Sign up to their mailing list, keep an eye on their social media, create an account with the venue to save time logging in (though in general I hate the whole account thing, but for this I make an exception).
Check the venue's website periodically to see if anything's been added. While writing this post I've just spotted that there are two tickets to #Travesties on sale for a performance at 3.30pm on Sat 1st Oct. Pretty sure they weren't available the last time I checked the site.
Sometimes tickets might be held back from general sale because someone (eg friends of the production team or a celebrity) have expressed an interest in one or more tickets but then may not be able to go, so the tickets can go on general sale.
Looks like 2 seats have become available for Saturday 1 Oct's 3.30pm perf of #Travesties https://t.co/qbaxKNnT3s (rare as hen's teeth!) :) pic.twitter.com/OQ9bju7P5B— Jo Brodie (@JoBrodie) October 1, 2016