Several people have posted that 90˚ is used very often as the angle for X/Y stereo pickup with cardioids, and I can't disagree with that--it is a very common choice. I can even remember that when I bought my first pair of cardioids over 35 years ago, the instruction booklet suggested placing them head-to-head at an angle of 90˚.
Someone even wrote that this angle was "standard." That perception is completely understandable, though in reality there is no such standard. My point is that if there ever were to be such a standard (even if it was only meant as general advice), 90˚ would be a poor choice for it, because of two things which I mentioned before:
[1] This setup results in a stereophonic recording angle of ±90˚, which is absurdly wide for general music recording. In particular situations that call for especially close miking of wide sound sources, or "all-around" pickup (e.g. discussion groups around a table), that approach has value, but for normal music recording with cardioids you usually want an SRA closer to ±50˚ or ±60˚. That requires either a larger angle between the axes of the microphones, some spacing and/or baffling between them and/or a sharper directional pattern.
[2] As you know, you can't make effective X/Y recordings with omnidirectional or "wide cardioid" microphones; the cardioid is the weakest classical directional pattern that allows X/Y stereo recording to function effectively at all. Fully 50% of the signal coming out of any cardioid is the pressure component of the sound field, which is omnidirectional. Thus with two coincident cardioids, even if you aimed them back to back (!), 50% of their output would be identical to each other (i.e. mono). Since no one uses nearly that wide an angle, the overlap in practice is substantially greater than 50%. This degree of correlation between channels is excessively high, and as a direct result, a good sense of spaciousness is not conveyed by such recordings.
If you want to follow the classical notion that the patterns of two microphones in an X/Y pair should intersect at their -3 dB point (i.e. where each one is delivering half power), then the "ideal angle" between an X/Y pair of cardioids would be ca. 131˚! This figure surprised me when I first encountered it in an AES publication, but mathematically it is correct; if anyone wants to discuss the formulas involved, we can do that.
The problem is that many real-world microphones, especially large multi-pattern microphones with dual-diaphragm capsules, are really only cardioid in the midrange. They become more like a "wide cardioid" at low frequencies and more like a supercardioid at higher frequencies. If you make X/Y stereo recordings with this type of microphone, the result is almost mono on the bottom regardless of what angle you choose, while at high frequencies you are constrained by the narrowness of the pattern. Some very well known, very high-priced studio microphones have rather poor off-axis response at high frequencies and are simply not a good choice for X/Y stereo recording even though the microphones may sound wonderful when used separately.
Most folks here seem to be using smaller, single-diaphragm condenser microphones, and that choice makes great sense if coincident or near-coincident stereo recording is what you mainly want to do. But one of the first things I noticed when I landed here was that the stock formulas for microphone distances and angles were being dispensed willy-nilly without any regard for the context in which they might be used. Most of what's been posted here (and not just here--on most Internet discussion groups about recording) is looking at only half the picture, and missing the basic relationship that you want to establish with your pattern choice and the geometry of your setup.
Let me suggest a different way of framing the issue. Most stereo recordings are intended for playback on a pair of loudspeakers that are some distance apart, with the listener some distance from the two loudspeakers. So the two loudspeakers and the listener form a triangle--often something fairly close to an equilateral triangle. From the listener's perspective, the various sound sources in the recording should seem as if they come from various angles within the left-to-right spread of the loudspeakers (and various distances, too--the illusion of depth which a good stereo recording can have). In other words, you want the individual acoustic sources to be "localizable"--perceptually you want them to "map" to points in the playback sound field which more or less resemble where they were coming from when the recording was made.
Coincident ("X/Y") recording can be very good at this mapping--much better than, for example, spaced omni ("A/B") recording. But you have to choose a directional pattern and a mike setup which more or less match the angular spread of the original sound sources with the angular spread of the playback environment, and that's why I mentioned the "stereophonic recording angle" earlier (to use Prof. Michael Williams' term for it--his book and/or his AES papers are very, very helpful in understanding all this). Cardioids at 90˚ would be terrific if stereo systems were typically set up with the left and right speakers on opposite (side) walls of the listening room with the listeners placed between them. If you've made X/Y recordings with cardioids at such a narrow angle, I suggest that you try moving your speakers well apart some time as an experiment--then aim them at each other and stand between them. It should greatly improve the stereo effect, since that is the type of playback system which really corresponds to an X/Y recording with cardioids at 90˚.
But if you're recording for a more normal playback environment, then for general-purpose music recording you really need to angle your cardioids distinctly farther apart than 90˚, and/or (as I said) choose a narrower pattern of microphone, and/or put some space or a baffle between the microphones to reduce the stereophonic recording angle of your setup. Otherwise you have to place the microphones so close to the sound sources that you will get huge imbalances, simply because the microphones will be proportionally so much closer to the sound sources near the front and center than they are to anything else.
Does that make some sense, for those of you who may have struggled through this whole rant?
--best regards