Suggested by Paolo Aluffi:

W. Fulton: Intersection Theory, Chapter 4 (for Segre classes)
R. MacPherson: Chern classes for singular algebraic varieties Ann. of Math 100 (1974), 423-432
G. Kennedy: MacPherson's Chern classes of singular algebraic varieties Comm. in Alg. 18 (1990) 2821-2839
A. Parusinski, P. Pragacz: Characteristic classes of hypersurfaces and characteristic cycles J. Algebraic Geom. 10 (2001) 63-79

Suggested by Joerg Schuermann:

"Besides the bibliography already suggested by Paolo Aluffi, I would especially recommend the following papers in relation to my approach to the theory of Lagrangian cycles."

(1) Sabbah,C.: Quelques remarques sur la ge'ome'trie des espaces conormaux. Ast. 130, 161-192 (1985)
(2) Grinberg,M.,MacPherson,R.: Euler characteristics and Lagrangian intersections. In: "Symplectic geometry and topology", IAS/Park City Math. Ser. 7, Amer. Math. Soc. Providence, 265-293 (1999)
(3) Goresky,M.,MacPherson,R.: Stratified Morse theory. Springer (1988)
(4) Schmid,W.,Vilonen,K.: Characteristic cycles for constructible sheaves. Inv. Math. 124, 451-502 (1996)

"Here I have of course ommited related work in the language of D-Modules and constructible sheaves (like Ginsburg, Kashiwara, Schapira, ...)."
"With respect to the theory of constructible functions, one can also look at:"

(5) Fulton,W.,MacPherson,R.: Categorical framework for study of singular spaces. Memoirs of the AMS 243, chapter 6 (1981)
(6) McCrory,C.,Parusinski,A.: Algebraically constructible functions. Ann. Sci. Ecole Norm. Sup. 30, 527- 552 (1997)
(7) Schapira,P.: Operations on constructible functions. J. Pure Appl. Algebra 72, 83-93 (1991)
(8) Fu,J.,McCrory,C.: Stiefel-Whitney classes and the conormal cycle of a singular variety. Trans. Amer. Math. Soc. 349, 809-835 (1997)

"I have also put some of my preprints on the arXiv-server."

Other references:

P. Aluffi: Singular schemes of hypersurfaces, Duke Math. J. 80 (1995), 325-351
J.-P. Brasselet: From Chern classes to Milnor classes. A history of characteristic classes for singular varieties. Advanced Studies in Pure Math. 29, (2000), pp. 31-52
T. Suwa: Indices of vector fields and residues of singular holomorphic foliations. Actualites Mathematiques. Hermann, Paris, 1998.