Many people have one alarm clock sound that wakes them up every morning.
Assume the alarm clock is going off at the same time every morning.
A general model of processing stimuli suggests that when information does not provide informational value, then we gradually begin to ignore it. Such a model is consistent with the experience of many people in relation to background traffic noise when moving from a quiet to a noisy neighbourhood. I.e., the frequency with which external traffic noise enters into personal awareness gradually decreases over time.
It seems that there is quite a lot of research on the effect of traffic noise on quality of sleep and the role of habituation (see this google scholar search on "habituation to noise sleep").
Kawada (1995) provides a review of the effect of traffic noise on sleep. Specifically in relation to habituation the author wrote:
Habituation to noise: Views and opinions on this parameter differ. Thiessen et al. reported habituation to intermittent wakefulness10～12. In their study, subjects were exposed to passing truck noise less than 20 times a night with a peak level of 65 dBA. The probability of a shift to shallower sleep and the amount of deep sleep demonstrated failure of adaptation over 24 successive nights. The probability of awakening decreased by almost one half in two weeks. Griefahn also studied habituation of 36 students to high-density road traffic noise and detected an all-night effect of noise on sleep for 12 consecutive nights13. Deep sleep was prolonged and shallow sleep including intermittent wakefulness became progressively shorter at 37 to 63.5 dBA. Vallet at al. indicate that there is no habituation to noise14. According to experiments on subjects living in a noisy environment for more than four years, young subjects showed less deep sleep and older subjects, less REM sleep under conditions of silence. Both Di Nisi et al. and Ohrstrom reported that there seems to be no habituation to noise exposure15,16. They exposed subjects to randomly distributed noise and it was impossible to predict either the time of exposure or the type of noise presented. Kawada and Suzuki could find no definite habituation to a noisy environment17. Their interval of noise exposure was 15 minutes and there was only one pattern of passing truck noise. Essentially the same noise characteristics are reported by Thiessen, but the findings differed from each other. Based on inter-individual variation in sensitivity to noise, habituation should differ according to the person.
I find this summary a little confusing. The between subject studies are not especially relevant in that habituation should presumably reduce the disruption caused by noise, but not necessarily remove the effect. Thus, it seems like longitudinal evidence would be the most relevant.
Muzet (2007) provides a relatively recent review of research on the effects of environmental noise on sleep. With regards to habituation the author wrote:
[Autonomic responses, such as heart rate changes and vasoconstrictions in response to noise while sleeping] have bee found not to habituate over long exposure times compared with clear subjective habituation over successive noise-exposed nights.
I found an article by Moorcroft et al (1997) that discussed the role of alarm clocks in peoples awakening process
The ability to awaken oneself from sleep at a preselected time without external means (such as alarm clocks) was studied using, first, subjective and, second, objective methods. First, in a telephone survey of 269 unselected adults, over one-half said that they never use an alarm clock (or other external means) or always awaken before it. Another 24% said that they sometimes awaken before the alarm. Furthermore, this ability positively correlated with age and was related to consistency in the amount of nightly sleep but not consistency in wake-up time. Second, 15 people who said they regularly self-awaken were objectively tested for this ability in their own beds using actigraphy for three consecutive nights while choosing their own wake-up times. Five awoke within 10 minutes of their target time (mostly before) on each night, five did so on two of the three nights, and of the remaining five, four did so on one night. Choice of target times varied considerably within subjects but more so for those who were more successful. Taken together these results show that many people have the ability to regularly awaken themselves from sleep at a desired time and that such an ability is of practicable utility.
I haven't found any specific empirical data on your question.
Habituation processes would suggest that a consistent noise that is deemed to carry a lack of informational content will gradually be more frequently ignored, and thus an inconsistent noise may be more effective in generating attention and perhaps wakefulness. That said, alarm clocks do carry informational content, and people could potentially tune themselves to that noise as the important noise to indicate the time to wake up.