After rocky, stumbling start, Carter administration aides have quickly learned which congressional power-brokers must be consulted and stroked before they dare announce second- and third-level political appointments in government.

A list of key Senate and House leaders is regularly checked by White House talent scouts - who briefly operated as free, if politically naive, spirits - to see whose blessing is necessary before candidates are selected for policy jobs.

Special attention is paid to getting clearance - or at least a pledge of benign indifference - in the House particularly where Republican appointees to special boards, commissions and minority seats in agencies are concerned.

Out of all this, House Speaker Thomas P. O'Neill Jr. (D-Mass.) has emerged as something of a jolly version of the godfather in political appointments. Once aloof Carter aides now "run by", at O'Neills convenience, people proposed for middle and top-level appointments.

The White House team ruffled O'Neill earlier with some appointments that were a surprise to him. They learned quickly that O'Neill doesn't like that kind of surprise, and that he has the clout to make life tough for every legislative proposal Carter has in mind.

"The Senate may have advise and constant powers," a much-burned Carter aide said yesterday "but O'Neill is the man to see, and clear things with if we want peace in the House."

One man who was run by recently was Chattanooga shopping center developer Joel W. Solomon, rumored to be in line to take over the government's housekeeping and purchasing arm, the General Services Administration.

"The Speaker knew Mr. Solomon slightly," a Capitol Hill source said, "and he saw him the other day in connection with the GSA job. He said he's OK! I don't know whether he'll get the job or not."

With the new White House recognition of O'Neill as the man to check with, his approval of Solomon is almost as good as press release announcing his nomination. Insiders who are trying to figure out who will get the job in government are watching the doors to O'Neill office and/or wishing they could get a peek at his appointment book.

Environmental Protection Agency administrator-designate Douglas Michael Costle is expected to get bad reviews from the EPA union this morning as the Senate Public Works Committee begins confirmation hearings.

Costle, a Californian with Capitol Hill work credentials, was with EPA as a consultant in mid-1975 before joining the Congressional Budget Office. The American Federation of Government Employees local of EPA (which wanted former Iowa Sen. Harold Huges to get the job) will question Costle's dedication to the merit system. It also wants to know what will be the status of current EPA appointees who recommended Costel for the consultant job, now that he is about to become thei boss.