The stereotypical view of the Doctor/companion relationship, one that RTD seems fully signed up to, is the Doctor travelling around with one attractive female. However, this situation only really existed for 12 out of the 26 seasons of the old show, and if you decide K9 counts as a full companion (I don't, personally) that goes down to eight. The reason this is the popular view is that it's the situation that obtained during the show's '70s heyday, and the "for the Dads" element. But if you ask me, it's all about family, and the different roles within the family that the Doctor and companions play at different times.
Disclaimer: Anything I say about attractiveness or shippiness is not meant to deny the fannish validity of anyone's favourite Doctor or pairing. I'm talking about what I see as the intentional role information canon gives us, rather than the subtext (which is where all the fun is). However hot some of us may find him, Six isn't portrayed as particularly attractive.
So going through the show's history, the different setups are:
Grandad, Grandad, you're lovely
One's era starts with the Doctor travelling with biological(-ish, if you're a die-hard Lungbarrow fan) family - Susan. Despite some initial spikiness, Ian and Barbara very quickly become the surrogate parents in this setup. Once Susan leaves, Vicki fills her role. Incidentally, this era is not short on romance within the family context. By the standards of '60s telly, Ian and Barbara are a canon ship. IMO, they're the OTP of all 44 years of Who. (Except for Doctor/TARDIS, but God/Alien Machine can be hard to relate to.)
After Ian and Barbara leave, we're down from three generations to two (through two-and-a-half: Steven starts off as a replacement Ian, and is clearly Older Than Vicki even if she is old enough to get a Married departure. I'd argue he's more equal with Dodo, give or take standard '60s sexism). The Doctor becomes the mad uncle you go away with for amazing holidays. This remains the case throughout Two's reign.
Three's era is the crucial transition point. In many ways, Three-in-exile is the black sheep uncle being brought back into the fold and made to do something useful for a change. The "younger relatives" of the family (Jo much more so than Liz, admittedly) are a bit in awe while the others like and respect him but also find him exasperating. The key difference here is Liz and Jo's status as the Doctor's "assistants" (which at one time was a word just as likely to be used by the media etc. as "companion") being different to that of the Brig, Benton, Yates et al - this (particularly with Jo) is where the idea that "there's the Doctor and one girl" really starts, even though at this point it's still embedded in a wider context that fandom readily admits is a "family". When Three gets away from Earth, it's only Jo who goes with him, though she never stays away for long and fairly quickly heads off with Cliff. (The "he's a young version of the Doctor" thing is played very heavily, but I don't think canon supports a "therefore she was doing it with him too" interpretation so much as "some people consciously-or-unconsciously looking for a parent figure in their mate".)
Single white females
Sarah is the first of the companions whose relationship with the Doctor is mostly portrayed in the one-to-one "girlfriend" style, though the transition away from the family-style UNIT era continues throughout her run, from the Earthbound parts of Season 11 through Harry's brief stint in the TARDIS to UNIT's last appearance until the '80s in Seeds of Doom. But most of the time it's just her and the Doctor, and of course for all that the romantic take on this in School Reunion can be seen as a retcon, it's a retcon by fiat - there's no attempt to sell us on the idea, it's just stated as fact, repeatedly ("the Mrs and the ex", etc.). I think you can see it there in canon, in part because a combination of SJS's adulthood and on-again-off-again-depending-on-the-writ
The rest of Four's era proceeds in one female companion mode. Moreover, each of them has some area of competency that make them more than just hostages for the villains - SJS's journalism, Leela's fighting skills and Romana's all round Time Lord winliness. Now, most companions in the past had something going for them too, but the important contrast is with Jo, who was explicitly coded as not much good at anything (except the escapology that fuelled the escape-recapture loops so beloved of overlong Pertwee stories) and who is many ways the prototype "one female" companion.
Leela's relationship with the Doctor is a bit more of a mentoring thing, but she is (as Talons points out) Eliza Doolittle to Four's Henry Higgins, and we all know how that worked out in the end. The Doctor's relationship with Romana is also mentory in a way (she leaves to be him, let's not forget), but it's also hella couply, and of course it gave rise to Who's very own canon RPF in the Tom'n'Lalla trainwreck. Overall, I think it's fair to say that we can read Four's relationships with his companions in the boyfriend/girlfriend mode, even if it doesn't come with the lashings of emo that that would imply on modern TV.
It's worth noting that the companions here are increasingly rootless. SJS doesn't seem to have strong links to Earth, though she returns to her journalism career when back on Earth in Seasons 11 and 12; Leela has been explicitly rejected by her tribe; and Romana never seems too concerned about returning to Gallifrey even in her first incarnation. We've moved away from a "family" family to a couple-based scenario.
Five's era sees a transition back to a more family-orientated mode with the biggest crew since Ben and Polly left. But this time, it's a squabbling, divided-against-itself family full of misfits and orphans. Five's TARDIS is Jim Robinson's house in spaaaaaaace, and not just because Tegan's Australian. This gets very interesting because Five is the most conventionally attractive of the first seven Doctors, and is coded as much more vulnerable than most (in other stunning news, the Pope may be Catholic). However, the era acts against canon shipping because of the "no hanky panky in the TARDIS" rule, however much Peter Davison may have been playing it as "he's shagging Nyssa".
Aces are rare
With Peri, Mel and Ace, we return to the Four-ish formula - one, relatively rootless, female companion and a tone that veers betwen coupley (though again, "no hanky panky" fights against it a bit) and mentor-ish. Like RTD now, JNT seems to be working to a formula of "giving the public what they think the show is". Peri and Mel should have various competences (Mel's computer fu especially) that never really come through on screen and Mel in particular is famous for being a screamer. Ace, OTOH, is introduced under Cartmel (worship him! worship him!) and is much more of a Leela type.
Now, in the same way as I think we can read Four/SJS, Four/Leela and especially Four/Romana into canon, I think you can make arguments for canon shipping here, even given the mentoring angle that's particularly evident with Ace. (At which point my inner ten-year-old-who-liked-Battlefield-best starts screaming "Nooooooo!" but we'll leave that on one side.) Put it this way - if it had been Ace in School Reunion, we'd still have had a retroship.
Who is that handsome madman?
The TVM takes us a step further with OMG the Doctor's first onscreen kiss (the first, it now seems, of many). The real difference in the TVM is that the Doctor is now coded as sexually attractive. He's always been coded as mysterious-stranger-who-you'd-love-to-go-t
On a more superficial level, Eight looks good. In another example of feeding the show's stereotypical perception back into the production, Eight dresses like an Edwardian, but in contrast to Three's eccentric dandiness and Five's apparent inability to ever change out of his sports kit, he's a well-dressed Edwardian. He is no longer just the hero, or the head of the TARDIS family, he's the male romantic lead. This is partly a consequence of modern relationship-centred TV's expectations (misidentified by fandom at the time as "American" rather than "modern"), but it certainly carries through into the intrepretation of the character in other media - both Sam in the books and Charley in the audios are proto-Roses, falling fast and hard for Eight.
The TVM also marks the beginning of a POV shift that "Rose" will complete. Although in this case the story starts from the Doctor's perspective, Grace is the identification character for non-fans. As the female lead, she's automatically going to get to snog the male romantic lead - in contrast, the fact that we haven't had a completely unambiguous Doctor/Rose situation on screen shows how restrained the new show has been about the shipping. Although, interestingly, like Donna at Christmas, she
Meet the family
And now we get to the 21st century series. There are two big things that I think make this era different from the previous "one female companion" eras. The first, and lesser, is that both Nine and Ten are not just coded as attractive the way Eight was, but are also flirtatious. Whether you're an asexualist or not, the previous Doctors were certainly fairly clueless on the romantic front, from One accidentally getting engaged to Cameca to Eight's "Oh, that was nice, do it again".
But the really big thing is that we're actually back to a family mode. The difference is that, as part of the POV shift to Rose as identification figure for newbies, it's no longer about the companion becoming part of the Doctor's surrogate TARDIS family, but the Doctor being introduced to the companion's real family. In contrast to the more-rootless-than-ever post-Time-War Doctor, Rose is more rooted than any previous companion, returning again and again to the Powell Estate, for all that she claims that travelling with the Doctor is what she wants most. In the end, it's not so much that the companion has become "the Doctor's girlfriend" but that the Doctor has become "the companion's boyfriend".
Look to the future now, it's only just begun
As for the future, the transition back to the Doctor as viewpoint character in S2 and the Runaway Bride means that I think we're going to get something closer to the Four and Six/Seven/Eight eras with Martha, except that the characters are Allowed To Talk About Their Feelings Onscreen now and so the canon shipping will be more overt. Certainly, "Made of Steel" makes me think she's going to be more rootless than Rose (she seems estranged from family and the book is her first return to the hospital).
 Now that I think of it, was this perhaps a reaction by JNT to the problems caused by the RL Tom and Lalla relationship?
 I wanted to write a bit about the exmaination of Ace's roots in S26, but I can't stop it from turning into a hair dye joke that wouldn't even make sense.
 Which I think has been successful for most of the new fans, but possibly not all. Have there been any "Rose has left, therefore I'm not interested" strops I've missed by not reading the right comms?