How much control do Time Lords have over the form of their next regeneration?

Several Doctors have made comments that show they have no conscious control over regeneration.

The Fourth Doctor examined his new face in the mirror and said "have to take the rough with the smooth". When the Eleventh Doctor met the Tenth Doctor in The Day of The Doctor the latter compared regeneration to a lottery. The Thirteenth Doctor showed surprise when she saw her reflection and said "oh, brilliant!" Nearly all Doctors have examined their physical characteristics in some way after regeneration. The Sisterhood of Karn also told the Eighth Doctor in Night of The Doctor that Timelord regeneration didn't have to be "random" - as their own regeneration science was more controlled.

However, other comments have demonstrated that there may be some subconscious influence over the outcome. In particular, the Twelfth Doctor subconsciously took the face of someone he had met before.

In Deep Breath, just after regeneration, he said:

TWELVE: Why this one? Why did I choose this face? It's like I'm trying to tell myself something. Like I'm trying to make a point. But what is so important that I can't just tell myself what I'm thinking?

He later realises, in The Girl Who Died:

TWELVE: I know where I got this face, and I know what it's for.
CLARA: Okay, what's it for?
TWELVE: To remind me. To hold me to the mark. I'm the Doctor, and I save people.

Evidently, this regeneration wasn't random, but entirely a subconscious choice. There is also the inference that he took an older face to clarify his relationship with Clara ("I'm not your boyfriend").

Arguably, the Sixth Doctor's personality may have been similarly chosen on a subconscious level. After regeneration, he commented that the change had come "not a moment too soon", and went on to say that he didn't like his previous generation's "feckless charm".

There was also the inference in Day of The Doctor that Ten and Eleven may have been younger and more child-like (in personality as much as physical appearance) as a way of not dealing with the horrors of the decision they made in the Time War ("Do you have to talk like children? What is it that makes you so ashamed of being a grown-up?")

So it would seem there is some control over the process, but it is at a subconscious level rather than a conscious decision. Having said that, we have seen a future incarnation of The Doctor in the form of 'The Curator', who said that it was possible to revisit "favourite" faces, but the fact he had to explain this to Eleven suggests that this is a privilege he did not yet have.

Quite a lot actually

While the Doctor demonstrates only subconscious control over his regenerations, the other Time-lords and Ladies we see seem to have rather more influence.

There is for example an extended sequence in Classic Who where Romana tries half a dozen different bodies before settling on the form of Lalla Ward.
It's played as a "Women and Clothes" joke, but the implication is that not only do you not have to be Near Death, but you can make very deliberate choices about the form you take.

This was long before the idea of there only being 12 regenerations came about. (not that that idea stuck when it came time for 13)