Collecting taxes from Americans has become an increasingly vexing problem for the Internal Revenue Service on this tax day, because:

(a) Some people cheat;

(b) Times are hard and a record number of people and businesses are not paying the money they owe;

(c) People with high incomes have developed an infatuation with tax shelters that are both poor investments and abuses of the spirit if not the letter of the tax code.

The problems are such that the Internal Revenue Service, unlike most agencies around town, is growing. If President Reagan's budget proposals for the IRS take shape, about 88,000 federal employes, a record, will be working for the tax collector by the end of fiscal 1983. More than 28,000 of them (up 5,000 from the end of fiscal 1980) will be involved in examining tax returns and 18,000 (up 7,000) will be trying to collect overdue accounts.

Thus, while IRS is cutting back on little taxpayer services like the toll-free telephone answer man, it is beefing up enforcement as more and more dollars escape the tax system.

First, the cheaters. There has been much press recently about the so-called "underground economy," where everybody does business in hard-to-trace cash, declines to report it on the Form 1040 and thus avoids both income tax and Social Security payments.

IRS Commissioner Roscoe L. Egger Jr. told Congress recently that the government is losing between $87 billion and $89 billion annually in taxes it would be collecting if everyone were totally honest. Those working on the federal budget would find their task much easier if they had an extra $87 billion or so to play with.

Then there's the hard times. On New Year's Day, the IRS books showed 1.6 million delinquent accounts. A delinquent account is a person, family or business that is more than three months behind in paying the tax bill, and those accounts owe the government $5.5 billion.

The problem is approaching epidemic proportions because only three months earlier, on Oct. 1 last year, there were 1.4 million delinquent accounts owing $4.7 billion. In other words, the amount owed on delinquent accounts jumped $800 million in just three months.

Half of those accounts were businesses, and almost three-fourths of their delinquencies involved either income taxes withheld from employes' paychecks or employer-employe contributions to Social Security that the businesses had not forwarded to the government.

"It's a serious problem," said IRS spokesman Ellen Murphy. "Employers file with us four times a year, and in three months they can get in trouble, especially when things are pretty tight." The IRS is trying to find ways, and possibly seek legislation, that will permit it to learn of an employer's difficulty soon enough that the problem can be solved without applying the ultimate remedy: closing the business and seizing the assets. Such action means, of course, that people lose jobs.

IRS officials view the increase in tax-delinquent accounts as a reflection of the economy in general. If a business gets tight and there's some money lying around in accounts being collected for the government, it is tempting to borrow today and swear to pay it back tomorrow.

Finally, tax shelters. There's is nothing inherently wrong under federal law with the concept of sheltering some of one's income from taxation--the Individual Retirement Account (IRA), for instance, has been blessed by both Congress and the president.

There is, however, another kind of tax shelter, a kind the IRS has taken to calling "abusive." Abusive tax shelters are playthings of the wealthy, usually appearing in tax returns of people who make more than $50,000 a year. An abusive tax shelter, loosely defined, is an investment designed to create losses the taxpayer can claim as deductions without having to risk much money.

An example: a taxpayer buys a $200,000 interest in an original lithographic plate and a limited number of prints. The actual cash invested, however, is only $30,000; the rest is obtained with a note, an IOU, that is to be paid only from the profits of the sale of the prints produced from the plate. If there are no profits, $170,000 worth of losses is created for a $30,000 investment.

The IRS under the Carter administration decided to challenge in court almost all squirrelly tax shelters. The result was that the Tax Court gagged on the litigation, with more than 8,500 shelter cases docketed by the end of last fiscal year. At the same time, more than 250,000 returns with tax shelters were being examined by IRS officials.

In an attempt to clear the backlog, Commissioner Egger has adopted a new policy of settling out of court as many cases as possible, and, although statistics are not yet available, IRS officials feel they are making progress and collecting money at the same time.

One thing helping the IRS close shelter cases and collect some past-due taxes is the fact that, since February, money owed the IRS carries an interest rate of 20 percent instead of 12 percent. When the penalty was only 12 percent, it made sense to keep litigation alive or to file a return but withhold the check, because where else can you get a 12 percent loan these days?