How does a budding artist get a big-city showing? There aren't enough galleries to care for all the painters who yearn for a New York exhibition, but here are some ideas that might work.

With roughly 600 galleries in New York, it may seem odd that so many artists cannot find a dealer to represent them. But consider: According to Whitney Museum estimates, 60,000 artists live and work in New York City alone. In addition, it is reasonable to assume that at least half of the other roughly 140,000 artists living throughout the United States view a show in a New York gallery as either a career necessity or a valued boost to their local reputations.

But will they get one? The statistics indicate that it's unlikely. And even if they do, it will probably be a one-shot deal in an undistinguished gallery (or a ``vanity'' gallery that can be rented for a few weeks), which will do little except credit them with having had a New York show.

How then, does an artist get his or her work shown if the regular gallery route appears blocked?

By personal contact with as many art professionals as possible; by continuing to show slides (or the work itself) to any dealer, curator, critic, or collector who shows the slightest interest; and by exhibiting whenever and wherever one can - even if the locations are less then ideal.

The latter isn't always easy. Pride may prevent one from showing in bank lobbies, restaurants, bookstores, corporate offices, or street fairs. And yet, some fair-to-good things can be found in these places. The work may be more promising than accomplished and unlikely to bring its creators immediate gallery contracts, but it's as good a way as any for talent to be spotted - and, they hope, remembered.

Cooperative galleries run by artists have also been helpful to many beginners. New members are voted in on the basis of their work. Upon acceptance, and the payment of monthly or annual dues, they are guaranteed at least one show every two years and inclusion in several group exhibitions.

Although cooperatives tend to be more open minded and far less commercially oriented than regular galleries, they have one great disadvantage: Critics and curators generally avoid them. Even so, they serve a valuable function in exposing talent, especially in the case of non-mainstream and beginning artists.

An even more successful method of showing work - at least in terms of sales - entails renting space in the Washington Square Outdoor Art Exhibit held twice a year on the sidewalks of Greenwich Village.

After an initial screening by a committee chosen from various professional art associations and the payment of $150, every artist and craftsman is assigned a 4-by-10-foot exhibition area for three consecutive weekends. Should that prove too small, additional units can be rented at the same rate. All exhibitors - there usually are between 300 and 350 - are automatically eligible for prizes.

Because of its age and reputation (it originated in 1931 and is now one of the best-known outdoor art-fairs in the world), the Washington Square Exhibit does quite well for its participants. Sales are relatively brisk, prices can often be negotiated on the spot, and commissions can be arranged.

The only problem, according to its critics, is the committee's acceptance of almost every conceivable type of art and craft, from the serious and accomplished to the frivolous and purely decorative. The exhibit's image, they insist, suffers as a result. Tourists may love it, but art professionals and important collectors tend to stay away.

Frivolity is certainly not in evidence in the exhibitions sponsored by Leger de Main, a not-for-profit, loosely cooperative organization of artists and collectors founded in 1984 to bring the work of talented newcomers to the attention of New York's Upper West Side. Its four exhibition spaces (plans for others are in the works), include the lobby of a residential hotel that has seen better days (Art Lobby, 2345 Broadway), and the legislative district offices of three local politicians (Ledo, 486 Amsterdam Avenue). Its 35 members exhibit through one of another of the galleries and pay modest annual dues.

Leger de Main, which is governed by a board of directors consisting of collectors, community leaders, and the directors of the four galleries, also sponsors annual new-talent exhibitions and publishes information and provides services for local artists. It exhibits the work of roughly 60 artists in over 30 shows every season.

The two other galleries are Outer Space, 2710 Broadway, and Steve Bush, 260 West 86 Street.

Leger de Main may be better organized than most, but it is by no means the only collective effort of its sort. Increasingly, artists are opening their studios once or twice a year as part of well-publicized community campaigns to focus attention on the art produced in particular neighborhoods. These events usually take the form of ``Art Tours,'' which include visits to a dozen or so lofts or studios and possibly an informal discussion.

Groups of artists are also banding together to rent vacant stores at reduced rates for the period it takes the landlords to find permanent tenants. This may be for only two or three months, but at least the work will have been seen.