If you are new to mysteries, there are lots of suggestions about where to start. A list compiled by the distinguished British author and critic H.R.F. Keating is a few years old but still serves as a good introduction:
Tales of Mystery and Imagination, Edgar Allan Poe (1845)
The Moonstone,
Wilkie Collins (1868)
The Mystery of Edwin Drood, Charles Dickens (1870)
Adventures of Sherlock Holmes, Arthur Conan Doyle (1892)
The Amateur Cracksman,
E.W. Hornung (1899)
The Hound of the Baskervilles, Arthur Conan Doyle (1902)
The Thinking Machine,
Jacques Futrelle (1907)
The Circular Staircase,
Mary Roberts Rinehart (1908)
The Innocence of Father Brown, G.K. Chesterton (1911)
Uncle Abner,
Melville Davisson Post (1918)
The Mind of Mr. J. G. Reeder, Edgar Wallace (1925)
The Murder of Roger Ackroyd, Agatha Christie (1926)
Red Harvest, Dashiell Hammett (1929)
Death of My Aunt, C.H.B. Kitchin (1929)
The Documents in the Case, Dorothy L. Sayers (1930)
The Maltese Falcon,
Dashiell Hammett (1930)
The Sands of Windee,
Arthur Upfield (1931)
Before the Fact,
Francis Iles (1932)
The Case of the Sulky Girl, Erle Stanley Gardner (1933)
Murder on the Orient Express, Agatha Christie (1934)
The Postman Always Rings Twice, James M. Cain (1934)
The Nine Tailors,
Dorothy L. Sayers (1934)
The Hollow Man,
John Dickson Carr (1935)
The League of Frightened Men, Rex Stout (1935)
The Wheel Spins,
Ethel Una White (1936)
The Beast Must Die,
Nicholas Blake (1938)
The Bride Wore Black,
Cornell Woolrich (1940)
Surfeit of Lampreys, Ngaio March (1940)
Calamity Town,
Ellery Queen (1942)
Tragedy at Law,
Cyril Hare (1942)
The High Window,
Raymond Chandler (1942)
Green for Danger,
Christina Brand (1944)
Appleby's End,
Michael Innes (1945)
Murder Among Friends,
Elizabeth Ferrars (1946)
The Horizontal Man,
Helen Eustis (1946)
The Moving Toyshop,
Edmund Crispin (1946)
The Fabulous Clipjoint,
Frederic Brown (1947)
The Franchise Affair,
Josephine Tey (1948)
Devil Take the Blue-Tail Fly, John Franklin Bardin (1948)
More Work for the Undertaker, Margery Allingham (1948)
My Friend Maigret,
Georges Simenon (1949)
The Asphalt Jungle,
W.R. Burnett (1949)
Smallbone Deceased,
Michael Gilbert (1950)
The Stain on the Snow,
Georges Simenon (1950)
The Daughter of Time, Josephine Tey (1951)
Last Seen Wearing..., HillaryWaugh (1952)
The Tiger in the Smoke,
Margery Allingham (1952)
Five Roundabouts to Heaven, John Bingham (1953)
The Long Goodbye,
Raymond Chandler (1953)
Post Mortem,
Guy Gullingford (1953)
The Party at No. 5,
Shelley Smith (1954)
The Talented Mr. Ripley,
Patricia Highsmith (1955)
Beast in View,
Margaret Millar (1955)
Gideon's Week,
John Creasey (1956)
Mystery Stories,
Stanley Ellin (1956)
Maigret in Court,
Georges Simenon (1960)
The New Sonia Wayward, Michael Innes (1960)
Gun Before Butter,
Nicolas Freeling (1963; also known as Question of Loyalty)
The Expendable Man,
Dorothy B. Hughes (1963)
Pop. 1280,
Jim Thompson (1964)
R.S.V.P. Murder, M.G. Eberhart (1965)
The Man Who Killed Himself, Julian Symons (1967)
Murder Against the Grain, Emma Lathen (1967)
Roseanne, Maj Sjowall &
Per Wahloo (1967)
The Last Best Friend,
George Sims (1967)
The Glass-Sided Ants' Nest, Peter Dickinson (1968)
Mr. Splitfoot,
Helen McCloy (1968)
The Private Wound,
Nicholas Blake (1968)
The Tremor of Forgery,
Patricia Highsmith (1969)
Blind Man With a Pistol,
Chester Himes (1969)
Young Man, I Think You're Dying, Joan Fleming (1970)
Beyond This Point Are Monsters, Margaret Millar (1970)
Sadie When She Died,
Ed McBain (1972)
The Friends of Eddie Coyle, George V. Higgins (1972)
The Players and the Game, Julian Symons (1972)
Mirror, Mirror on the Wall,
Stanley Ellin (1972)
The Dance Hall of the Dead, Tony Hillerman (1973)
The Poison Oracle,
Peter Dickinson (1974)
Fletch,
Gregory McDonald (1974)
The Black Tower, P.D. James (1975)
The Long Shadow,
Celia Fremlin (1975)
The Naked Nuns,
Colin Watson (1975)
The Blue Hammer,
Ross MacDonald (1976)
Sleeping Murder, Agatha Christie (1976)
A Death in the Life,
Dorothy Salisbury Davis (1976)
The Investigation,
Dorothy Uhnak (1976)
A Judgement in Stone,
Ruth Rendell (1977)
Laidlaw,
William Mcllvanney (1977)
Nobody's Perfect,
Donald Westlake (1977)
A Pinch of Snuff,
Reginald Hill (1978)
Skinflick,
Joseph Hansen (1979)
Kill Claudio,
P.M. Hubbard (1979)
The Green Ripper,
John D. McDonald (1979)
All On a Summer's Day,
John Wainwright (1979)
Death in a Tenured Position, Amanda Cross (1981)
The Glitter Dome,
Joseph Wambaugh (1981)
To Make a Killing,
June Thomson (1982; also known as A Portrait of Lilith)
The False Inspector Dew,
Peter Lovesey (1982)
The Artful Egg,
James McClure (1984)
A Taste for Death, P.D. James (1986)