As someone from outside the field of mental illness you may first wish to understand that psychiatric disorders do not exactly approximate the "hard definitions" of other sciences. Criteria for specific disorders have changed over time and particular disorders may come and go. A good discussion of "what is a mental disorder" is attached.

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3101504/

There is no simple answer to this question. Perhaps you get some ideas from the references to some papers.

Al-Hamzawi, Ali Obaid; Rosellini, Anthony J.; Lindberg, Marrena; Petukhova, Maria; Kessler, Ronald C.; et al. (2014). The role of common mental and physical disorders in days out of role in the Iraqi general population: Results from the WHO World Mental Health Surveys.

Journal of Psychiatric Research53 (Jun 2014): 23-29.

Kessler, Ronald C.; Demler, Olga; Frank, Richard G.; Olfson, Mark; Pincus, Harold Alan; et al. (2005). Prevalence and Treatment of Mental Disorders, 1990 to 2003.

The New England Journal of Medicine352.24 (Jun 2005): 2515-2523.

MacKenzie, Thomas D.; Kolpak, Steven J.; Mehler, Philip S.. (2005). Prevalence and Treatment of Mental Disorders: Comment.The New England Journal of Medicine353.11 (Sep 2005): 1184.

Subramaniam, M.; Abdin, E.; Vaingankar, J. A.; Verma, S.; Chong, S. A.. (2014). Latent structure of psychosis in the general population: Results from the Singapore Mental Health Study.Psychological Medicine44.1 (Jan 2014): 51-60.

Llosa, Augusto E.; Ghantous, Zeina; Souza, Renato; Forgione, Fabio; Bastin, Pierre; et al. (2014). Mental disorders, disability and treatment gap in a protracted refugee setting. The British Journal of Psychiatry204.3 (Mar 2014): 208-213.

Sandra Flynn et al. (2014). Serious Violence by People With Mental Illness

National Clinical Survey

This study aimed to estimate the prevalence of mental disorder in offenders convicted of serious violence, examine their social and clinical characteristics, and compare them with patients convicted of homicide.

Narayan, Angela J.; Allen, Timothy A.; Cullen, Kathryn R.; Klimes‐Dougan, Bonnie. (2013). Disturbances in reality testing as markers of risk in offspring of parents with bipolar disorder: A systematic review from a developmental psychopathology perspective.

Bipolar Disorders15.7 (Nov 2013): 723-740.

Oram, Siân; Flynn, Sandra Marie; Shaw, Jenny; Appleby, Louis; Howard, Louise Michele. (2013). Mental illness and domestic homicide: A population-based descriptive study.

Psychiatric Services64.10 (Oct 1, 2013): 1006-1011.

Very tough question Jeff.

I don’t think there is a definitive answer because it depends on many factors. Rates across studies will vary greatly according to some of the following considerations, just to name a few major ones. Some may account for the changes in rates of mental illness over time.

- how mental illness is defined (e.g., official diagnosis, number of symptoms present, a general ‘severe mental illness’ category)

- how mental illness is measured (e.g., self-report/survey, standardized interview)

- the particular disorder (e.g., schizophrenia, mood, bipolar, personality disorders, developmental disorders, etc)

- type of prevalence (e.g., point, lifetime, incidence)

- the population under question (e.g., forensic, civil psychiatric, general community samples, also important variations due to age, gender, ethnicity)

- the classification system used (e.g., ICD-10 vs the many versions of DSM)

- date of the study (older vs newer)

To my knowledge, there is no definitive gold standard study for world wide rates. The WHO study mentioned in a previous answer might be a good start. You can find some good estimates if you focus on a specific region or population. 

I agree this is a tricky topic due to the difficulty studying large populations over time use the same methodology. You might have luck search for "temporal trends" and "national survey" or similar. Also there seems to be some data on distress and depression but little on schizophrenia and bipolar.

See these potentially useful references:

J Affect Disord. 2015 Mar 15;174:556-61. doi: 10.1016/j.jad.2014.12.039. Epub 2014 Dec 18.

Trends in psychological distress, depressive episodes and mental health treatment-seeking in the United States: 2001-2012.

Mojtabai R1, Jorm AF2.

Can J Psychiatry. 2015 Jan; 60(1): 31–34.

 The Prevalence of Major Depression Is Not Changing

Scott B Patten et al

BMC Psychiatry. 2015; 15: 193.

An evaluation of variation in published estimates of schizophrenia prevalence from 1990─2013: a systematic literature review

Jason C. Simeone et al