How many classification of mathematical topics exists? I found only one Mathematics Subject Classification, are there more? 
 A: Even if you consider only the MSC, you'll get more than one classification system if you wait long enough.  The MSC gets updated every ten years or so.
A: Yes, there are more ways of classifying the subject areas of mathematics; classification schemes are merely human constructs that attempt to impose some sort of organization or structure to the domains of mathematics in a manner that is useful and that makes sense:
The MSC (*Mathematical Subject Classificatio*n) is a fairly standard scheme. Here is a little history and overview of the MSC. And what follows are some suggestions for 


*

*$(1)$ different representations of the MSC, AND 

*$(2)$ different (alternative) classification schemes of organizing the different branches of math.


$(1)$ See the Mathematical Atlas (www.math-atlas.org), for a "Math Map" of subject areas in mathematics which depicts the interconnections of the various branches in math. 
The webpage with the "Math Map" is searchable, and you'll find the linked page "Subject Index". (Click here: "Browse: list of subject headings", or you can click that same link at the bottom Math Map* webpage linked above.)
Top-level subject areas in the MSC, each of which is "searchable" through the Mathematical Atlas, if you want to explore "sub-levels" encompassed by "top-level" subjects:
00: General material including elementary mathematics
01: History and biography
03: Mathematical logic and foundations
05: Combinatorics and graph theory
06: Order, lattices, ordered algebraic structures
08: General algebraic systems
11: Number theory
12: Field theory and polynomials
13: Commutative rings and algebras
14: Algebraic geometry
15: Linear and multilinear algebra; matrix theory
16: Associative rings and algebras
17: Nonassociative rings and algebras
18: Category theory, homological algebra
19: K-theory
20: Group theory and generalizations
22: Topological groups, Lie groups
26: Real functions and elementary calculus
28: Measure and integration
30: Functions of a complex variable
31: Potential theory
32: Several complex variables and analytic spaces
33: Special functions including trigonometric functions
34: Ordinary differential equations
35: Partial differential equations
37: Dynamical systems and ergodic theory
39: Difference and functional equations
40: Sequences, series, summability
41: Approximations and expansions
42: Fourier analysis
43: Abstract harmonic analysis
44: Integral transforms, operational calculus
45: Integral equations
46: Functional analysis
47: Operator theory
49: Calculus of variations and optimal control; optimization
51: Geometry, including classic Euclidean geometry
52: Convex and discrete geometry
53: Differential geometry
54: General topology
55: Algebraic topology
57: Manifolds and cell complexes
58: Global analysis, analysis on manifolds
60: Probability theory and stochastic processes
62: Statistics
65: Numerical analysis
68: Computer science
70: Mechanics of particles and systems
74: Mechanics of deformable solids
76: Fluid mechanics
78: Optics, electromagnetic theory
80: Classical thermodynamics, heat transfer
81: Quantum Theory
82: Statistical mechanics, structure of matter
83: Relativity and gravitational theory
85: Astronomy and astrophysics
86: Geophysics
90: Operations research, mathematical programming
91: Game theory, economics, social and behavioral sciences
92: Biology and other natural sciences
93: Systems theory; control
94: Information and communication, circuits
97: Mathematics education 


$(2)$ And for Alternate classifications of mathematical subjects, see: 


*

*Library of Congress 

*Dewey Decimal 

*Referativnyi Zhurnal [125Kb] 

*NSF Mathematics Programs

*"Encyclopedic Dictionary" system 

*ArXiv Preprint server 

*MAA Basic Library List 

*Historical systems (1900, 1930)

*still more alternate classifications... 

