In most parts of most photographs, photon shot noise is the largest contributor to noise.

Mostly, we're comparing it to read noise. (Dark current is negligible in short exposures, and quantization noise is also pretty small when you're talking about 12- and 14-bit ADCs.) Read noise depends on the sensor. This 2007 paper presents read noise measurements for a few DSLRs. We see, for example, that a Canon 40D at ISO 200 has about 10 electrons (e-) of read noise.

Photon shot noise is a Poisson process, so the noise is the square root of the count of signal photoelectrons. So if we record 100 signal photoelectrons in a pixel from our subject, we expect the shot noise per pixel to be sqrt(100)=10 e-, equal to the 40D's read noise.

Is 100 photoelectrons a lot? No, the same paper estimates the full-well capacity of a 40D pixel to be 56,000 e-, so a pixel with only 100 e- is a very dark part of the scene, about 9 stops darker than full-well. In a pixel with more than 100 e-, the shot noise continues to increase, up to sqrt(56000)=236 at full-well, so the shot noise dominates the read noise by a larger and larger margin. (The bright tones appear less noisy than the dark tones, because the signal-to-noise ratio continues to increase, as the noise is only the square root of the signal. But what noise there is, is due increasingly to shot noise, not read noise.)

In the very dark shadows, the read noise may be significant. And in a long, dark exposure (such as astrophotography under dark skies), dark current and read noise may both be important. But for general photography of well-exposed subjects with short exposure times, shot noise is the dominant source of noise.