Sequential Placement Optimization Games
Todd Neller
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Sequential Placement Optimization Games share these common features:

Each turn:
* a game piece is randomly drawn, and
* each player places the piece on his/her individual game board (generally not to be moved from its fixed position)
At the game end, each player scores desired piece configurations achieved on his/her individual game board.

Take it Easy!, Word Squares, and Poker Squares are examples of such games. In alternation, common cards/tiles/letters are drawn and announced and player(s) commit to a fixed placement of such cards/tiles/letters in a grid in order to create scoring combinations/sequences along grid lines. Once the grid is filled, player(s) sum individual grid line scores to compute a total final score. All such games can be played as either solitaire or competitive games.

If you know of other games that share these characteristics, please add them below.

Reiner Knizia described a Sequential Placement Optimization Game of his design called Criss-Cross (and variants) in his book Dice Games Properly Explained (pp. 182-187). This was said to have been published in 1992. According to the Convivium Knizia ludography (http://www.convivium.org.uk/pdfs/Web-Games-20030906.pdf), Criss-Cross was published in Games & Puzzles magazine in the first issue of 1995. BGG currently has no record of the game in its database. If you have a copy and can verify publication details, please create the BGG entry and let me know so as to include it here.

Notes:
* Games like Carcasonne and Oceania have players building on a common board. Here, players seek to individually optimize their placements.
* Although the examples above score along lines, in general scoring configurations of any sort would fit this category.
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1. Board Game: Karuba [Average Rating:7.32 Overall Rank:364]
Todd Neller
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Grid: square
Placing: tiles with paths
Scoring: movement of figures along paths to collect tile bonuses and reach destinations faster
 
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2. Board Game: FITS [Average Rating:6.61 Overall Rank:1164]
Stephen Roney
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Ladera Ranch
California
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Each board has different rules for scoring the placement.

Scoring is not exactly along grid lines, but otherwise this fits.
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3. Board Game: Take it Easy! [Average Rating:6.60 Overall Rank:1318]
Todd Neller
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Grid: hex
Placing: tiles with colored lines
Scoring: unbroken lines of single color
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4. Board Game: Cities [Average Rating:6.41 Overall Rank:2063]
Dave Lartigue
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Springfield
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This game fits your criteria. It is also way boring.
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5. Board Game: Take it to the Limit! [Average Rating:6.84 Overall Rank:2698]
Alfred Wallace
United States
Champaign
IL
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The successor to Take it Easy! It shares the same qualities.
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6. Board Game: BITS [Average Rating:6.47 Overall Rank:3274]
Stephen Roney
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again, the scoring is not strictly along grid lines, but each player is playing the same pieces in the same order with the same goals (except for a starting piece difference)
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7. Board Game: Don Quixote [Average Rating:6.06 Overall Rank:3894]
 
Dave Lartigue
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If I remember correctly, this game is sort of the opposite of what you're looking for, yet related. In it, it's the location that's the same for all players, but the pieces they can place in that location are different.
 
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8. Board Game: Streams [Average Rating:6.39 Overall Rank:4623]
Brian M
United States
Thornton
Colorado
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Streams has players placing numbers along a "stream". Numbers are drawn one at a time, and each player places the number. You ideally want to get long chains of ever-increasing numbers.
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9. Board Game: Take it Higher! [Average Rating:6.82 Overall Rank:5175]
Alfred Wallace
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Then there's this, which I haven't played. The BGG description:

"This game takes "Take it Easy!" as a basis and adds three new elements. First of all the tiles are octagonal, with four scoring lines running through them. Secondly, the center of the tiles are either silver or gold giving you additional bonus points once you score a colored line where all the tile centers are silver or gold (9/10 bonus points per tile). Finally, when you complete a colored line, you might be able to put a rocket ship of the corresponding color on the tile you just placed. These rockets are available from the start of the game on numbered circles on the individual playing boards. At the end of the game you will receive bonus points for the highest empty space that has no rocket ships lower than it."
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10. Board Game: Word Squares [Average Rating:6.81 Unranked]
Todd Neller
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Grid: square
Placing: letters
Scoring: words according to length
 
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11. Board Game: Poker Squares [Average Rating:4.58 Unranked]
Todd Neller
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Grid: square
Placing: playing cards
Scoring: poker hands
 
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12. Board Game: "Not Another One!" [Average Rating:5.62 Unranked]
Daniel Ajoy
Ecuador
Quito
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