Minimum safe distance for black holes

Fortunately (or unfortunately) depending on your outlook, Frontier did not model black hole behavior accurately. You can approach it just like any other stellar object(that's not a star) and be pretty OK.

to prove this, several CMDRs flew through a black hole.

As Rapitor has noted in their answer, it is currently safe to approach most black holes even to the point of reaching the "Body Exclusion Zone" if they have one.

Personally, I've made this a bit of a hobby now. Whenever I find a black hole that's in a place with an interesting view, I'll go "faceplant" it and get as close as I can.

Different ships handle heat differently, so be mindful of your own ship's tolerance for this. However, most black holes do not emanate much heat if any.

The process for approaching a black hole is rather simple, and similar to entering other unmarked locations of interest such as planetary rings without a Resource Extraction Site. Just approach it slowly, keeping an eye on your ship's speed and heat levels. Make sure the ship isn't accumulating heat faster than it can tolerate, and get your speed under 1 Mm/s before you're forced out of Supercruise.

After the emergency drop from Supercruise though, is where black holes can get a little weird. Here, you run into one of two cases:

1. The black hole has a Body Exclusion Zone properly implemented.

You'll be able to fly towards the black hole for a little while. Probably just a couple boost's worth. Then, suddenly, your ship will stop moving towards it. Even with full throttle and boost, you won't be able to get any closer and your HUD will show an alert "Body Exclusion Zone Hit".

Most black holes I've encountered have this at around 20-40 km. However, there have been some with Body Exclusion Zones greater than 100 km. Then, there's the rare and really big ones like at the Great Annihilator - I don't particularly recall how big that one was when I got there, but I do remember it was bigger than any others I'd seen.

2. The black hole does not have a properly-implemented Body Exclusion Zone.

These are relatively few and far between, and the game developers may be working to fix them. For these black holes, either the Body Exclusion Zone just doesn't exist or it doesn't actually block ships from entering it - I'm not sure which.

What will happen with these, is what's shown in the videos linked by Rapitor. You can effectively fly all the way through the center of these black holes, and the effects are rather spectacular.

IMPORTANT EXCEPTION - SAGITTARIUS A*

Do not attempt a faceplant at the Supermassive Black Hole!

My Asp Explorer generally runs pretty cool. I can typically get that ship out of a faceplant with any star or other object, without taking heat damage and without using heatsinks, and I can sit pretty deep in the corona of most stars for a good few minutes without substantial risk.

But for Sagittarius A*, faceplanting is not an option for that ship. As far out as 40 Ls, the heat levels began to rise rather quickly and I had to abort because I didn't have the spare heatsinks to risk going further.

I might consider trying it again some time, if I build a ship that runs much cooler and packs a lot of heatsinks. If I do though, I will be going in with the full expectation that it is a suicide mission. (Though I'll try to remember to change this answer if I survive!)

WARNING - GREAT ANNIHILATOR

Be careful here!

The black holes at Great Annihilator are absolutely amazing. There are few others like them in the galaxy, and their size makes for some rather spectacular visual effects - especially on your initial entry to the system.

I highly recommend a visit to Great Annihilator if you're ever anywhere near it, but be careful if you want to attempt a faceplant at either of the black holes there. I personally have hit the Body Exclusion Zone on both, but I did notice that the space around them was a bit warmer than most other black holes I've encountered.

Mind your heat levels as you approach these, as you should for any stellar body - including other black holes. If you've got a ship that runs cool enough, you should be able to go in just fine. If your ship can't take the heat though, get it out of the kitchen!