Probably the best way to handle this for me is to go chronologically
Maha Go Go Go/Speed Racer- very vaguely Michi/Trixie and Go/Speed (I was only 4 1/2-5 years old when I saw the series (1971) so my shipping was pretty vague at that age. But I vaguely remember liking it when Trixie hugged Speed. )
Beatle related- John and Yoko (my first true shipping, and only real life one)
Gatchaman/Battle of the Planets- Ken/Mark and Jun/Princess (kind of vague shipping, I sort of halfway also liked Jason/Princess)
Star Wars- Han Solo and Princess Leia
Elfquest- Redlance and Nightfall
Yamato/Star Blazers- Susumu/Derek and Yuki/Nova
Logan's Run- Logan and Jessica (TV series and novel versions, characters are too dull in the movie version)
Macross- Max and Miriya (also it's 1990's sequel Macross 7)
Bubblegum Crisis (has to be a prize winner for worst/most non fitting series title ever. Good OVA series- horrid name

- Leon and Priss
Maison Ikkoku- Yuusaku and Kyoko
City Hunter- Ryo and Kaori
Ranma 1/2- Ranma and Akane, and although not developed past an initial stage- Ryouga and Ukiyo (the two least insane rival characters for Akane and Ranma. A quite popular pairing amongst the western fandom, but Takahashi hinted at two late introduced characters as the probable real pairings for Ryouga and Ukiyo- which probably disappointed fans of the series about as much as Rowling not having Luna and Neville get together in Harry Potter...) my sister was a big fan of Ryouga and Ukiyo (and also Luna and Neville in Harry Potter. I could see the plausibility of the matches and admit a slight disappointment when they didn't happen, but instead late series characters ended up with them or in Luna's case, a totally unseen character...)
Back to roughly chronological order now
Rose of Versailles- (my first shoujo series

Andre and Lady Oscar
Titans- Jericho and Raven
Cloak and Dagger- Tyrone and Tandy
Justice League- Guy and Ice
Diane Tregarde series- Andre and Diane
X-men- Gambit and Rogue
Anita Blake series- Jean-Claude and Anita
Sailor Moon- Haruka and Michiru
Rayearth- Ferio and Fuu, Umi and Ascot
NGE- Misato and Ryoji (only one of the teenage characters I liked was Asuka so no ships there for me. Didn't particularly like the series either...)
Himechan no Ribbon- Daichi and Himeko
Hana Yori Dango- Domyoji and Tsukushi (but only after Domyoji really fell for her and got goofy about it, not his initial psychologically messed up attraction)
Kodomo Omocha- Akito and Sana
Fushigi Yuugi- Miaka and Tamahome/Taka (liked his Taka incarnation better), Tokaki and Subaru
X-Files- Mulder and Scully
Kenshin- Kenshin and Kaoru
Card Captor Sakura- Lee and Sakura (don't care for the Tsubasa alternate worlds series- Princess Sakura is
way too passive, and basically just a damsel in distress- ugh.)
Ayashi no Ceres- Tooya and Aya
Kamakaze Kaitou Jeanne- Chiaki and Maron
Alice 19th- Frey and Mei-Lin (hinted at toward the end of the manga. This was the first Watase manga I found the male romantic lead- Kyo, pretty uninteresting and ditto the female romantic lead Alice. I like the support characters better- especially Frey and Mei-Lin)
Buffy The Vampire Slayer- Spike and Buffy
Harry Potter- Snape and Lilly (Rowling said it might have happened had the two of them made different choices at crucial times.)
Inuyasha- Miroku and Sango, Inuyasha and Kagome
Full Metal Alchemist- Roy and Riza, Ed and Winry
Shugo Chara- Ikuto and Amu (although the age difference bothered me when they were kids- Ikuto's 16 to Amu's 10- the main story ends just as Amu is turning 13, which makes Ikuto 19, but thankfully Peach-Pit ends the series separating Ikuto and Amu for a few years before things get too disturbingly sexy between them.) It's clear from Tsukasa's last words to Amu in the original Japanese manga (Viz's translation muddles things- maybe because of Ikuto's and Amu's age difference- maybe afraid if an American parent saw that they might write angry letters) that Ikuto and Amu will end up together, and Tadase will, as an adult, disappear from the future timeline to go back into the past- taking the name Tsukasa (and the assumed identity as Tadase's uncle) and act to protect Ikuto and the Guardians- including his younger self and Amu.
Dr. Who- The Doctor and Rose. The other kind of permanent Doctor pairing with Riversong is just too flaky for me. If it wasn't canon it would look seriously cracked