I have seen the three words frequently used to express similar meanings. But what exactly is their difference that tells me which is the most suitable one in a certain case?
For the most part, they are used in different contexts.
Field is most often used to designate a domain of professional specialization: you would speak of an attorney in the field of patent law, a doctor in the field of pediatrics, or a scholar in the field of Middle English literature. You might say that someone that is in the legal or medical or LitCrit field if you had only a vague idea of what specialization he pursued—“He’s something in the medical field”—but you would not do so if you had already implied that larger domain: ✲“He’s a surgeon in the medical field”. You would not, except for humorous effect, use the term with a non-professional: a ✲“machinist in the aeronautics field”, a ✲“secretary in the legal field”, a ✲“merchant in the medical-supply field” (though you might speak of an “aeronautical machinist”, a “legal secretary”, a “medical-supply merchant”).
This invidious distinction is not observed with industry, which designates a domain of commercial endeavour and embraces para-professionals, businessowners, managers, technicians, skilled labourers, support staff, and unskilled workers as well as professionals. It is typically used to designate the “supply chain' for particular classes of product: the steel industry, the energy industry, the entertainment industry, the education industry, the financial-services industry. The term is mostly used statistically and is not ordinarily applied to individuals, except in relation to industry-wide conditions: “Joe's in the construction industry and having a hard time finding work“ is OK, but not “Joe's a manager in the construction industry”.
Sector is similar to industry but is usually employed to distinguish domains on a structural rather than input/output basis: the public and private sectors, the non-profit and for-profit sectors, the manufacturing and service sectors. Its use is almost entirely statistical; you rarely speak of an individual as being in this or that sector, except when you speak of an individual moving from one sector to another, as when a defeated congressman becomes an industry lobbyist.
None of these distinctions is hard-and-fast. An insurance agent, for instance, may claim professional status by describing himself as working in the field of personal protection. Industry may come in handy in a bio or resume to describe an individual's varied career: “I have held responsible positions in every aspect of the housing industry”. Sector may replace industry when speaking narrowly of a particular aspect in which different domains are compared: for instance, an analyst may contrast the performance of the construction, manufacturing and service sectors on the stockmarket or in creating jobs.
In terms of business/economic discussion (especially in investing) sector is a broader term than industry. According to Investopedia [emphasis added]:
When breaking down the economy, the first groups are sectors which describe a general economic activity. Then all of the companies that fall into that sector are categorized further into industries where they are grouped only with companies with which they share very similar business activities.
Sectors describe broad categories within the overall economy:
Dividing an economy into different like-pieces allows for more in-depth analysis of the economy as a whole. Any economy can be divided into sectors, such as the economy of a particular city, or the global economy. The oil and gas sector is an example of an economic sector.
However, this is not a hard and fast rule especially when the words are used independently, in which case they are often synonymous. Likewise, the term "industry sector" can also be used to refer to broader sectors.
Industry is often used (in the United States) as a formal means of classifying occupations and employment. The Bureau of Labor Statistics (BLS) defines industry as:
A group of establishments that produce similar products or provide similar services. For example, all establishments that manufacture automobiles are in the same industry. A given industry, or even a particular establishment in that industry, might have employees in dozens of occupations. The North American Industry Classification System (NAICS) groups similar establishments into industries. NAICS is replacing the former Standard Industrial Classification (SIC) system.
While "industry" and "sector" are often used in discussing economics, "field" is a broader term and not necessarily related to any sort of enterprise. The OED for American English gives the definition for field as:
a particular branch of study or sphere of activity or interest