You make an assumption that the non-rainy-funds are relatively illiquid.

What's common is to have a targeted mix of assets, often simplified to Stocks/Bonds (Yes, one can get very fancy, Stocks/Bonds/Metals/RealEstate/Crypto. For this example, let's keep my response simple).

Say my mix is Stocks 70% / Bonds (note, bonds include CDs, money market funds etc) 30%. When I need cash, it naturally comes from the Bond/Cash assets. This at least delays the need to pick what to sell from the stock side. That decision comes during times of reallocation, as I don't really perseverate over the 70% increasing to 75 or dropping to 65 as that's the result of a moving market in either direction. Of course if one has individual stocks and not just a broad index, those stocks are monitored for potential sale. To make it simple, if your allocation were so aggressive you have no bond/cash position, you'd revert to the advice of selling the stock that's not one you'd buy today, one that no longer looks desirable.

There is no definitive answer to this question which is why you are having trouble finding an answer with your search. A lot depends on your goals and investment style.

If you use an asset allocation model, which may include low cost index funds, you would sell those investments to keep your desired model. If your funds are out of balance, withdrawal could use to bring them back in line. Or, you would sell a small amount from each "bucket".

If you are a more active trader, seeking to maximize profits, you would sell those investments in which you deem less likely to be profitable in the future. How you determine that depends a lot upon how you determine your investments in the first place.

I recently had to do something similar myself. Since I am a asset allocation guy, I just sold off funds that preserved my asset allocation.

A good question to ask yourself about anything you own that could be sold is "Would I pay X to own that item today"? The X is what it could be sold for. If the answer is "no" then sell the thing.

Buying a car ...

Consider the list "threw away 10,000 on strippers", "gambled away 10,000 at the casino", "spent 10,000 on a car".

The first two are fine: (A) they are one-time expenses and (B) at least you have a good memory. But "spent 10,000 on a car" is financial insanity. Because the costs are ongoing, monthly, extremely substantial, and never-ending.

Buying a car is financial madness.

Selling stocks to buy a car is a really unfortunate idea.

It's hard to imagine anything more negative than selling stocks to buy a car.

Buying a house...

Money put in to buying a car is pretty much the single most sickening waste of money possible. In contrast money put in to buying a house is, in almost all cases, an entirely good thing. So, great.

Of a group of stocks, which to sell: tax implications

Regarding the question "of a group of stocks, which to sell", in many jurisdictions there are tax differences between your stocks A, B, C, D due to issues such as length of time held.

If this is the case, you have the answer.

Of a group of stocks, which to sell: trading decision

Unfortunately, deciding which to sell of A, B, C, D is exactly the same as deciding which stock to buy of a group of stocks.

It is a straight-up stock picking exercise.

You literally have to time travel to the future and decide which one will go-up-least in the future.

As you have noted,

"Non Answers - I googled around and there seems to be a plethora of advice..."

Any advice on which stock to sell - or buy - from a group A, B, C, D is: bullshit. Total bullshit.

The final, unequivocal answer is: you must guess which one will go-up-least in the future, and that's it. Any other advice is as valueless as stock-tip newsletters unfortunately.

Regarding measures such as "volatility". If "volatility", or anything else, was a predictor of the future price of a stock - everyone would be a trillionaire. Nothing, at all, has predictive power of the future price of a stock; there is no stock picking.

One possible logical approach

People often ask "which stock to buy?" The only answer is "buy an index fund, because, obviously, it's completely impossible to stock pick."

Similarly, some would say that: if you have to cash out from stocks A, B, C, D. The only possible way to do this is to tell the future and know which one will go-up-least and sell that one. Hence, since stock picking is comic, the best thing to do is "sell the basket" .. equally sell from A, B, C, D.

In this way you minimize the risk of losses.

Speaking of an S&P index...

If one currently owns stocks A, B, C, D. It's better to sell them all and buy a plain broad index fund. One could take this opportunity to dump all of A B C D, take away the relevant cash for the house, strippers or gambling (forget the car: never sell stocks for a money-waste), and put the rest of the money in an index fund.

Step 1: Rebalance your portfolio with the cash subtracted

Rebalancing is something you SHOULD be doing once a year or so anyway. If you don't: insert Step 0: learn how to rebalance and do it regularly Subtract the desired cash-out and calculate how much you want to have in each asset class after you are done. Than you can calculate the difference between the target amount and what you currently have: that will give you the amounts to withdraw and/or invest for each asset class.

Step 2: Look at the tax implications

This can be complicated and is highly depending on your local tax laws. Example: In the US long term gains are taxed less than regular income but withdrawing from retirement accounts can be expensive. Figure out which accounts and specific holdings are tax friendly. In some cases, you may have to adjust your allocation target somewhat. That's not a problem: regular rebalancing will eventually get your there over time.

Step 3: Look at past performance

but don't over-rotate on it and don't try to "predict" where things are going. There are different school of thoughts on this but a reasonably careful approach would be "preferentially sell the stuff that has been doing well" . Most asset classes go through cycles: if you sell a strong asset you cash in the gains while given the weaker ones more time to get to the strong part of their cycle. Don't sweat it too much though: you will always win some and loose some and as long as you are well diversified, that should all average out.

Step 4: Execute

Once you have figured what and how much to sell, withdraw and re-invest, you just need to do it. It helps enormously to have a "good" brokerage account and investments that are easily movable. For example I have a brokerage account at Vanguard (no advertisement intended, but I can recommend it) holding a very diverse set of ETFs. I could liquidate the whole thing in about 10 minutes. No fees, no delays in the execution of the trades, no brokers to call, etc. Don't overthink the timing: today will be better than tomorrow for some funds and worse for others. You have no way of predicting this, so you might as well just get it over with.