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REVIEWS IN BRIEF *As Thousands Cheer: The Life of Irving Berlin Laurence Bergreen (Viking, $24.95) A colorful history of our greatest popular art-the music that emerged out of the ethnic and social commotion of 20th- century America-as well as a vivid biography of its foremost exponent. A
*Boone Brooks Hansen and Nick Davis (Summit, $19.95) An overly ambitious, desperately chic first novel about a performance artist and cult hero. C
*The Burden of Proof Scott Turow (Farrar, Straus & Giroux, $22.95) Although The Burden of Proof lacks the surefire thriller devices that helped make Presumed Innocent such a commanding piece of storytelling, readers who persevere will be rewarded. B
*Children of the Dragon: The Story of Tiananmen Square (Collier, $19.95) A handsome collection of photographs, speeches, and newspaper accounts documenting the ill-fated democracy movement in China. B+
*Coyote Waits Tony Hillerman (Harper & Row, $19.95) The 11th tale in Hillerman's series about the Navajo Tribal Police is sturdy work from an incorruptible craftsman. A-
*Crooning John Gregory Dunne (Simon & Schuster, $19.95) Essays full of high dudgeon and sometimes high reward. B
*Crosstown Traffic: Jimi Hendrix and the Rock 'n' Roll Revolution Charles Shaar Murray (St. Martin's, $18.95) The best book yet on Hendrix-and also a sweeping historical discussion of soul, jazz, the blues, and the impact of technology on pop music. A
*Get Shorty Elmore Leonard (Delacorte, $18.95) Hollywood has been good to Elmore Leonard, and Leonard has returned the favor with a blackly comic thriller set there. A-
*Hit Men: Power Brokers and Fast Money Inside the Music Business Fredric Dannen (Times Books, $19.95) A revealing look at the men who manipulate the rock & roll business. A
*The Innocent Ian McEwan (Doubleday, $18.95) Far more than an intelligent thriller, The Innocent is also a haunting black comedy charged with psychological complexity, sex, and suspense. A
*Jazz Singing: America's Great Voices From Bessie Smith to Bebop and Beyond Will Friedwald (Scribner's, $29.95) The author's quirks aside, this is quite simply the best available guide to the art of jazz singing. A-
*Language Maven Strikes Again William Sa re (Doubleday, $24.95) The sixth collection of Sa re's pun- mongering ''On Language'' columns. B
*Saratoga Hexameter Stephen Dobyns (Viking, $16.95) A charming small-town mystery-comedy. B+
*Seventh Heaven Alice Hoffman (Putnam, $18.95) A young divorcee sparks a mystifying outbreak of passion in a Long Island suburb. B
*Stardust Robert Parker (Putnam, $18.95) A new episode for Spenser, mysterydom's most self-satis ed shamus. B-
*Too Good to Be True: The Outlandish Story of Wedtech James Traub (Doubleday, $21.95) As Traub deftly shows us, this is a story not about venality but about human frailty. A
ALSO NOTED *Among Schoolchildren Tracy Kidder (Avon, paperback, $9.95) A closely observed, somewhat melodramatic, best-selling account of one teacher's classroom successes.
*A Dictionary of the Second World War Elizabeth-Anne Wheal, Stephen Pope, & James Taylor (Peter Bedrick Books, $29.95) From Aachen to Yugoslavia.
*Indian Style Suzanne Slesin and Stafford Cliff (Clarkson Potter, $45) Indian interiors and exteriors, from maharajahs' palaces to mud huts. With an introduction by Ismail Merchant and James Ivory.
*Journey Into Cyprus Colin Thubron (Atlantic Monthly Press, paperback, $9.95) Vignettes from a 600- mile trek across Cyprus.
*Memories of Midnight Sidney Sheldon (Morrow, $21.95) A sequel to The Other Side of Midnight, Sheldon's '74 best-seller.
*Seamus Heaney: Selected Poems 1966-1987 (Farrar, Straus & Giroux, $20) Generally considered the greatest Irish poet since Yeats, Heaney does not shrink from the murderous subjects of recent history.
