Does milk with different fat contents (skim, 2%, whole, etc.) spoil at different rates?
And when spoilage does occur, are milks with higher fat content more sour than those with less?
Are there smell differences?
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No, there is no reason they should.
Bacteria feed on carbohydrates, not on fats. (This is why oil doesn't spoil outside of the fridge - it is pure fat). So it is the amount of milk sugar which is important for the bacteria, and it is the same regardless of the fat content. Also, the spoiled milk is not more or less sour at the end.
The other important factors are initial bacteria count and storage temperature. You can't do anything about the bacteria count, and can't know what it is. The storage temperature is, of course, your fridge temperature, which should be in the small range of 0-4°C.
The other factor is the type of pasteurization. Traditional pasteurization makes a milk with a shelf life of 7 days, ESL gives it a shelf life of 21 days, and UHT milk can last for months, uncooled. Once opened, each of these types spoils within 3-4 days. And then, normally pasteurized milk goes really sour, partly even separating into curdles and whey. ESL and UHT milks get slightly more bitter, but not much, and they stay liquid and don't change their odour or colour. Unless you pay lots of attention to the taste, you can't tell when ESL milk is spoiled. But this happens the same way no matter the fat content.
I don't think that rumtscho's answer is quite correct.
I explained some of this in my answer to How does butter remain edible for so long without refrigeration?. Milk spoilage is caused principally by lactose content, and the presence of Lactobacillus bacteria feeding on lactose (or any other sugars, which is why sweetened condensed milk spoils so quickly).
Fat doesn't prevent Lactobacilli from doing their thing, but it's a simple question of ratios; all other things being equal, higher-fat dairy products have a lower lactose content. See for example this chart of lactose content in various foods. Low-fat milk has on average 5 g (per 100 g), full-fat milk has 4.8 g, half-cream has 3.3 g, double cream has 2.5 g, and so on all the way down to butter which has only 0.6 g.
There's not much of a difference between skim milk and full-fat milk in terms of lactose content, and in fact skim milk from one brand or batch may have less lactose than full-fat milk from a different brand or batch. This uncertainty is why all milks (but not creams, cheese, butter, etc.) tend to be lumped into a single spoilage category; however, on average full-fat milk has less lactose than non-fat and will spoil at a slightly slower rate.
Pasteurization and initial bacterial counts do, of course, have a more profound effect, but fat content does have an impact on the bacteria's "food" and therefore an overall impact on the spoilage rate, even if it's not easy to measure at home.
As far as sourness goes, fat goes rancid over time which leads to a sour taste, but that process happens much more slowly in milk than bacterial spoilage, so I would not expect full-fat milk to taste any sourer than skim milk at the same stage of spoilage.
My son and I did this for science fair project. We left milk in a glass out at room temp . Day 1 whole milk and 2% was about the same. On day 2 the whole milk had a very slight odor and a slight film over top of glass. The 2% had a stronger odor and had seperated it had an inch of curds on top and inch of liquid on bottom.