Beam Me Up, Scotty!: "Nul points", the phrase where a country gets no points is a popular term in the British populist media and Eurovision fans, but the phrase actually doesn't translate to "no points" in French (French for "no points" is actually pas de point and zéro point) and is not used as an official term in the contest or is announced by the presenters. Terry Wogan who co-hosted in 1998 and popularized the term had a golden chance to actually mention this when Switzerland got "nul points", but this was not to be. The closest thing to this happened in the 2016 contest when changes to the voting system meant that the Czech Republic got no points from the public vote. It was announced as "zero points".
Early-Bird Cameo: Some people will have participated for their country (or for a different country as backing vocals, a dancer, writer, composer, or musician in one year, and then return as the competing artist themselves.) Similarly, the inverse happens, an artist who competes one year will return in a reduced capacity. Sometimes they do this many times.
United Kingdom's Blue competed in 2011.
People like Cascada (Germany) and Bonnie Tyler (United Kingdom) competed in 2013. Anouk (The Netherlands) counts too, although she's more famous in her home-country than in the rest of Europe.
Inverted with ABBA (Winner 1974 Sweden) and CÚline Dion (Winner 1988 Switzerland), who got famous because of Eurovision.
In Memoriam: The opening to the 2015 final in Vienna included a short tribute to Udo Jürgens, Austria's first Eurovision winner (and the only one before Conchita), who died the previous December, by way of a violin rendition of his winning song, "Merci, chérie" ("Thank You, Darling").
Missing Episode: The 1964 ESC in Copenhagen has either been lost or shut away, depending on who you ask. The most prevalent theory is that the Danish broadcaster taped over or lost their only copy. Video of the winner's reprise exists, as does audio of the whole show. The first contest in Lugano (1956) is also lost, although there is newsreel footage of the winning song and an audio recording (with 20 minutes missing) also exists.
In 1999, Rosario "Chayo" Mohedano, niece of all-time great Rocío Jurado, was initially announced by many media outlets as the Spanish entrant with Lere lele, the lead single of her debut album Agua de sal. However, TVE backed down and went with Lydia in what developed into a no-win scenario in the long run: Lydia finished last in Jerusalem with one solitary point (and took home the Barbara Dex award as the worst dressed in the contest), while Mohedano's musical career tanked and she only found relevance thanks to longstanding Mohedano family drama.
Comedian Bill Bailey has considered entering for Britain; whether he has attempted or been knocked back or whether he hasn't gotten round to doing it yet is unknown. He is indeed a talented musician (he has absolute pitch, ie. he is able to identify a note just by its sound), but we still have to remember that he's known mostly for his surreal comedy and bizarre musical interludes during his show, including a tribute to the music in Starsky and Hutch. Still, considering they're unlikely to win in the future anyway, they've got nothing to lose. Bill Bailey himself said as much on his twitter page.
There was a persistent rumor in 2004 that the UK was going to try and do some damage control after Jemini's... less than stellar performance the previous year and send former Spice Girl Emma Bunton. Emma has said a few times that she'd like to do the show, but it's never came to be.
Conchita Wurst tried to represent Austria back in 2012, only to lose out to the Trackshittaz by a mere one percent of the vote. Of course, two years later, Austria decided to give her a chance, perhaps taking this trope to heart, and she went on to win it all.
Similarly, Jamala was one of the hopefuls to represent Ukraine in 2011, with the song "Smile", and placed third in the original final, but dropped out of the race due to the preselection's skewed voting process. Five years later, she decided to give it another go with "1944", and of course won it all in Stockholm.
That year Zlata Ognevich was another potential candidate with "The Kukushka". She placed second, in between Mika Newton and Jamala, but she too decided to withdraw from the new final. Just two years later she represented Ukraine in Malmö, and finished in a respectable third place, behind Azerbaijan and Denmark.
Swedish-Greek singer Helena Paparizou (winner for Greece in 2005) could have represented Sweden in 2014, but she was outvoted in favor of Sanna Nielsen, Ace Wilder and Alcazar.
And, once again, Loreen could have been representing Sweden in 2017, but got eliminated in Andra Chansen. Imagine the Broken Base and outcry this caused among European fans. On second thought, don't.
This video goes into detail of entries that were disqualified, withdrawn due to circumstances etc. at any rate, would have represented their respective countries but ended up not. These examples go back as far as 1967.