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I-187 (preliminary numbering) New York
On Aug. 22, 1958, the New York Department of Public Works submitted "Interstate 187" for the Westchester Expressway from Elmsford to White Plains. On Aug. 29, 1958, AASHO replied that this should be called I-487, since it connects two interstates (87 and 95). 
The I-487 moniker prevailed until 1961, when it was decided to consider it as part of I-287. 
I-287 New York; New Jersey
98.72 miles ; from I-95 in Port Chester, N. Y. around NYC, Paterson and Newark to I-95 near New Brunswick. Interstate 287 crosses the Hudson on the Tappan Zee Bridge. It's the third longest 3di in the U.S., second if you don't count I-476.
Already well underway in 1965, I-287 didn't see its last segment (I-80 to I-87) completed until November 1993, more than 25 years later. 
Early Numbering History
On Aug. 22, 1958, the New York Department of Public Works submitted a numbering plan that considered today's I-287 as a pair of spurs from I-87. From Elmsford to I-95 in White Plains would be I-187; from Elmsford to the New Jersey state line at Suffern would be I-387. On Aug. 29, 1958, AASHO replied that I-387 should be part of I-287, to match New Jersey's proposal; and I-187 should be called I-487, since it connects two interstates. 
In 1961, I-487 was redesignated as part of I-287. 
Once-proposed Long Island Sound Crossing
In 1966, Robert Moses proposed a $150 million, 6.1-mile bridge to extend I-287 to NY 135 (Seaford-Oyster Bay Expressway) on Long Island. Gov. Rockefeller endorsed the idea, but Suffolk and Nassau residents were strongly opposed. The fight continued for over seven years.
The final blows arrived in 1973:
More information: Oyster Bay-Rye Bridge (unbuilt) (Steve Anderson)
I-287 in New Jersey
Don't go yet, there's more. From William F. Yurasko:
"I-287 had the longest segment of incomplete Interstate road until Nov. 1993 when the section in Boonton (not far above I-80) was connected to NJ 17 in Mahwah. The entire 20.5 miles had been on the drawing board since the fifties and nothing happened. In 1985, the path was altered near Oakland, and shortly thereafter construction began. The section from NJ 17 to I-87 NY Thruway was rebuilt after the intial opening. Another note: Gov. Jim Florio had tried to have it opened the week before the election, to take credit (previous gov. was real force for it), but the Army Corps of Engineers delayed the opening for two weeks to conduct drainage tests. Florio lost. When driving up and down I-295, check out all the "IMPEACH FLORIO" stickers on exit signs."
Chris Blaney adds: "Now, I-287 in NJ, the first 45 miles of which were built in the early to mid 1960s, has new pavement [c. 1994] on about 55 of its 67 miles, is now six to ten lanes wide (except for the part north of Exit 59, Route 208, which is 4 lanes), has brand new exit signs with the proper milepost numbers (which need to be added to a lot more NJ highways), and features a smooth, comfortable ride at high speeds. (75-85 mph. no problem on even the busiest sections.) They've done a good job overall.
"The quad-carriageway section between Exit 17 (Bridgewater) and Exit 21 (Route 78) went from 2-2-2-2 to 3-3-3-3. A single left-shoulder HOV lane has been installed between Exit 17 and Exit 41. To accommodate this, the highway from Exit 22 (202-206, Bedminster) north to Exit 33 (Harter Rd.) was widened from 2 lanes each way to 3; through Morristown and to Exit 37 (Rt. 24 Freeway) the road was widened to 4 lanes each way; and from Exit 37 to Exit 41, the roadway is 5 lanes each way. There is an unusual "triple-carriageway" between MP 40.5 and 42. The 2 left lanes on the southbound side are separated from the other 3 by a Jersey wall. These left lanes come from Route 80 West. From Route 80 West, you cannot exit I-287 at Exit 40 because of the Jersey wall."
Even NANAE gets the occasional roadgeek thread
"... Watch for the combo "I-87 South / I-287 East" Also, I-287 in Jersey is basically an offshoot of I-87 (if you're willing to stretch the definition) and isn't a loop at all.-- Inigo Montoya, "Re: DHS October President's newsletter - Happy Harvesting!" Online posting, news.admin.net-abuse.email
I-387 (preliminary numbering) New York
On Aug. 22, 1958, the New York Department of Public Works submitted "Interstate 387" for the New York State Thruway between Elmsford and Suffern. On Aug. 29, 1958, AASHO replied that this should be called I-287 and connect to a proposed I-287 in New Jersey. 
I-487 (numbered as another interstate) New York
I-487 (cancelled) New York
In 1965, I-487 was the number for the proposed 47-mile Hudson River Expressway, running from I-87 in the Bronx along the Hudson's east bank to I-84 in Beacon (which at the time was also I-87; see I-684.
Within three weeks of the I-487 plans' unveiling, the section from the Bronx to Tarrytown (at I-287) was dropped. By mid-1968, this plan was reduced further to a 10.4-mile route from Tarrytown north to Ossining. Plans to use Federal funds were dropped due to mounting opposition from the Federal government, notably Interior Secretary Stewart Udall.  All that remains in the area today is the US 9 freeway from Ossining to Peekskill.
I-587 New York
1.21 miles . Too small for many maps, I-587 runs east from near I-87 into Kingston. It has no interchanges and does not actually connect with I-87; a traffic circle connects I-587's western terminus to I-87's exit ramps.
To I-87's credit (grin) it has no signs mentioning I-587; though as soon as you take the exit, there are signs for I-587 at the traffic circle. 
I-587's designation was proposed by the New York Department of Public Works on Aug. 22, 1958, and was approved by AASHO. 
For other short 3di's, see: Shortest and Longest 3di's.
See also: I-587 (Steve Anderson)
I-687 (cancelled) New York
Albany expressway plans in the 1950s included an east-west expressway from exit 5A on I-90 to a proposed exit 3 on I-87. Another expressway would continue west to Albany International Airport, and eventually to Schenectady.  The I-90 to I-87 road, proposed Interstate 687, was at times called the Northside-Northway Connection, or the Airport Connector. 
The Exit 5A interchange was built in 1971 along with the rest of I-90. No interchange was built on I-87, but a gap was left in the exit numbering to accommodate I-687.
In 1972, the Capital District Regional Planning Commission showed a longer I-687 plan, extending west of I-87 to the junction of an upgraded NY 155 and a proposed "Scotia Niskayuna Arterial."  A few years later, no highway was planned at all: opposition in the town of Colonie spurred I-687's deletion from the regional plan in 1973, and from the interstate system in 1977.  The money was eventually used for upgrading the I-87/I-90 interchange and possibly for a NY 7 upgrade.
Exit 5A was eventually used for Corporate Woods Blvd in Colonie; Exit 3 on the Northway doesn't exist, but is being studied (in 2002) for improved access to the airport. 
Gary G. Nelson writes: "I rather suspect [I-687] was just a strawman to get the tradein funds... eventually used for Alt. NY Route 7 and some other projects. I was involved in fighting the latter in the late 1970's." 
I-787 New York
10.16 miles ; north from I-87 in Albany to NY 7 outside Troy, where NY 787 continues as a surface road for 2 more miles. Then I-787 turns east over the Collar City Bridge with NY 7, and ends at 8th St. in Troy. 
Robert Philhower writes that the eastward turn of I-787 is on an exit ramp that appears to leave the mainline. It was well marked before NY 787 was constructed, but is now signed more subtly.
See also: I-787 (Chris Jordan)