Never appeared forthcoming papers This has been inspired by this MO question: Harmonic maps into compact Lie groups
Just for joking: which is your favourite never appeared forthcoming paper?
(do not hesitate to close this question if unappropriate)
 A: Here is a gap in a famous series of papers.
G.H. Hardy, and J.E Littlewood
Some problems in Partitio Numerorum,  VII
Their series of papers "Partitio Numerorum" is quite influential in the development of the 
Hardy-Littlewood circle method.
Some comments on the missing part are on page 253 in a paper by 
R.C. Vaughan, Hardy's legacy to number theory,
Journal of the Australian Mathematical Society (Series A) (1998), 
65: 238-266. Cambridge University Press
http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=4937088
A: EGA, Chapters 5 through 12
A: The Igusa-Waldhausen paper (roughly) entitled, 
The expansion space model for $Q(X_+)$
which is supposed to give a very different proof of the splitting $A(X) = Q(X_+) \times \text{Wh}^{\text{diff}}(X)$ that is based on a description of $Q(X_+)$ as the moduli space of finite relative cell complexes over $X$.
A: *

*S. Gel'fand, Yu. Manin, Methods of homological algebra, first appeared in Russian as Методы гомологической алгебры. Введение в теорию когомологий и производные категории. Т. 1 (that is VOLUME 1). Volume 2 has never appeared in Russian and also has been given up and the Springer Western edition does not cite Russian original, has many typing errors in formulas which Russian original does not have and it scraped off the tome 1 from the title. 

*M. Demazure, P. Gabriel, Groupes algebriques, tome 1, Mason and Cie, Paris 1970 -- later tomes/volumes never appeared

*Z. Semadeni, Banach Spaces of Continuous Functions, Polish Scientific Publishers, Warzawa, 1971, second volume never appeared from the Polish Sci. Publ. There is however a different book with a similar title in Springer in 1982, Schauder bases in Banach spaces of continuous functions. Lecture Notes in Mathematics 918. Springer 1982. v+136 pp. MR83g:46023.

*John W. Gray, Formal category theory: adjointness for 2-categories, Lecture Notes in Mathematics 391, Springer-Verlag 1974. xii+282 pp. has been envisioned as a m3 volume project on formal category theory, some material is mentioned in volume 1 and never appeared. The monograph is very innovative and some of the material from the latter volumes was undoubtfully sketched by the author in some detail. The author later drifted to theoretical computer science. 

*John Duskin started a paper in several parts "Nerves of bicategories", part I appeared with great delay as http://www.tac.mta.ca/tac/volumes/9/n10/n10.pdf, partly due serious health problem the author experienced few years ago. Second part "Bicategory morphisms and simplicial maps" and the promised third part did not appear, although the contents description looks very promising. We wish the author good health and more to be seen!

*H. Amann, Linear and Quasilinear Parabolic Problems, Volume II: Function Spaces and Linear Differential Operators. This second volume was cited as "in preparation" already in 1997 (see doi:10.1002/mana.3211860102) and continued to be cited this way even in 2016 (see doi:10.1007/s00028-016-0347-1). As of 2017, it seems the book still has not appeared. The structure has changed at least to some extent, too, so that what was meant to be the first chapter will no longer be included (it is made available for free from http://user.math.uzh.ch/amann/books.html instead).
Grothendieck planned not only later EGAs but also later SGA (e.g. some Berthelot's works in SGA 8).  Bourbaki Elements are of course never finished as well (an now are very slow, asymptotically stalling) as the German encyclopedic work by Klein's students at the beginning of the 20th century. M. M. Postnikov wrote two volumes of a course on algebraic topology in Russian about basics of homotopy theory and promised the homology in "next semester", but no books appeared on that. 
A: There is a result by Oesterle, that proves that you can find the first non residue quadratic modulo a prime in no more than $70\log(p)^2$ step assuming the GRH, this result was then improved by Bach who replaced the constant $70$ by $2$. The result of Oesterle was never published and when I asked him why, he told me because the laptop containing the proof was stolen from his car. However I think he exposed his proof to the mathematical community, so it is widely recognized. 
A: This one is famous. It has been at the origin of a huge mathematical activity (conservation laws, homogenization, weak KAM, Hamilton-Jacobi equations, etc ...):

P.-L. Lions, G. Papanicolaou, SRS Varadhan. Homogenization of Hamilton-Jacobi equations

A: J. Berge. Some knots with surgeries yielding lens spaces.
(c. 1990; cited by 92 on Google Scholar.)
A: Kurt Gödel referred to part II (Der wahre Grund für die Unvollständigkeit, welche allen formalen Systemen der Mathematik anhaftet, liegt, wie im II. Teil dieser Abhandlung gezeigt werden wird, darin, daß die Bildung immer höherer Typen sich ins Transfinite fortsetzen läßt) in his seminal paper Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte für Mathematik und Physik 38 (1931) p. 191. This part never appeared.
A: This doesn't exactly count as an unpublished forthcoming paper, but the supposed original proof of Fermat's Last Theorem that was "too large to fit in the margin" should probably be mentioned here.
A: B. Farb. Automorphisms of $F_n$ which act trivially on homology.
A: Volumes 4 through 7 of The Art Of Computer Programming. 
A: A. Bertrand-Mathis, Le $\theta$-shift sans peine
A: The comment about stacks in the paper that first used them in an essential way probably belongs in this list:
"Full details on the basic properties and theorems for algebraic stacks will be given elsewhere." (Deligne-Mumford, The irreducibility of the space of curves of given genus, 1969.)
They don't quite say they will give the details in a paper, of course, so maybe it doesn't count.
A: Steven Krantz tells the following story, Mathematical Apochrypha, page 136: 
My Ph.D. thesis was based in part on work of Walter Koppelman of the University of Pennsylvania. My source was a very brief research announcement that Koppelman had published in the Bulletin of the AMS. I could never find the promised subsequent paper that would fill in all the details, and I had to fill them in myself. I eventually went to my thesis advisor and asked him where the missing paper was. He said, "Oh, God. Don't you know?" And then he told me the sad story. There was a very unhappy graduate student at the University of Pennsylvania. He had had bad experiences with several thesis advisors (at least so he thought), the last being Koppelman. One day he went into the colloquium, shot the department chairman, shot Koppelman, and shot himself. Koppelman and the student died. 
There's a report of the story in Observer-Reporter - Feb 12, 1970, page 23 of 32:
https://news.google.com/newspapers?nid=2519&dat=19700212&id=WsddAAAAIBAJ&sjid=bV4NAAAAIBAJ&pg=906,1950274
A: W. Crawley-Boevey. The Deligne-Simpson problem.
A: "The Aarhus integral of rational homology 3-spheres IV," by Bar-Natan, Garoufalidis, Rozansky and D. Thurston, never appeared. I think developments in the field overtook the need for the paper, which was referred to in the first paper in the series. This is a great series of papers by the way. Very clearly written.
A: Dana Scott and Robert Solovay, "Boolean-valued models of set theory"
A: Gromov's seminal "Pseudo holomorphic curves in symplectic manifolds" (1985) refers 10 or 15 times (for explanations of further applications that he only refers to or sketches briefly and for even "further discussion on $\overline{\partial}_\nu$ for non-regular curves") to his forthcoming "Pseudo holomorphic curves in symplectic manifolds, II", listed as "in press" by Springer.
It never appeared.  Gromov wrote a few later papers on symplectic geometry, but never returned to holomorphic curves.  The paper is the foundation of modern symplectic topology (Floer homology, quantum cohomology, Gromov-Witten theory, symplectic field theory, etc.)
A: Nobody can compete with Fermat, but papers confidently labelled with the roman numeral I and never followed by II might fit here.   Of these my favorite is one by Tits, Normalisateurs de tores I in J. Algebra 4 (1966).
A: I'm a fan of Peter May's book The Homotopical Foundation of Algebraic Topology (feel free to correct the title if I've got it wrong).  It has been referred to by May in various places, and sounds really interesting!  But it has never been written.
A: How about "The classification of finite quasithin groups" by G. Mason from 1980? The classification of finite simple groups was announced when G. Mason was still working on this important case and he then abandoned the work. This hole in the classification was closed finally in 2004 by M. Aschbacher and S. D. Smith.
A: The sequel to Kontsevich's "Deformation quantization of Poisson manifolds, I" has never appeared.
A: Deligne's construction of the Galois representations attached to modular eigenforms (he did give a sketch in a Bourbaki talk though).
A: The books Classical Banach Spaces III and Classical Banach Spaces IV by Joram Lindenstrauss and Lior Tzafriri never appeared (after having been promised in various places of volumes I and II). 
As written by Albrecht Pietsch in his book History of Banach Spaces and Linear Operators, the reason the later volumes never appeared was that "the development was too vigorous. Thus, in order to finish this project, a complete rewriting would have been necessary". Even still, the influence of volumes I and II in Banach space theory has been exceedingly nontrivial; indeed, Pietsch also writes: "The two-volume treatise of Lindenstrauss/Tzafriri on Classical Banach Spaces has become the most important reference of the modern period".
A: Jeff Smith's book on combinatorial model categories.  
A: Kervaire, Milnor: Groups of homotopy spheres II.
In the introduction to part I, they write:
"More detailed information about these groups will be given in Part II.
For example, for $n = 1, 2, 3, \ldots, 18$, it will be shown that the order of
the group $\theta_n$ is respectively:" (a table follows). Similar remarks are scattered throughout the text.
The details have been written down by other people and it must be said that part I contains the much more complicated arguments.
