The classical collection of mostly real-life problems is at NETLIB. The files are compressed with a special utility (MPC), they need to be uncompressed using EMPS. Then they are in MPS-format. The original source for the files and the (un)compress utilities is netlib. See the README. One can also retrieve the files in AMPL-format from NETLIB_AMPL. A comprehensive archive for files in different formats is the COAP collection.

Here is another large collection of mpc-compressed MPS files and here and here are some additional files.

In practice often some or all of the variables are constrained to be integer valued. Here is a collection of mixed integer linear programming problems:

Testcases for MILP

MIPLIB 2017 (MPS format)
MILPlib (MPS format) MILPlib (AMPL format)
Coral MIPlib (MPS format)
Alper Atamtürk's Collection (MPS format)
UniBo (MPS format) difficult cases from U Bologna

Testcases for MIQP and MIQCP

QPlib (multiple formats) Both convex, non convex, continuous, discrete instances
MIQPlib (ext MPS format)
MIQPlib (AMPL format)

Stochastic Programming Testcases
in SMPS format

Sampling Methods for SP
POSTS (POrtable Stochastic programming Test Set)
Stochastic Integer Programming Test Problem Library
Another testset from various applications

Testcases for transportation problems

Fixed Charge Transportation Benchmark MPS format

Testcases for unconstrained minimization

Wikipedia's list some global test problems, also multiobjective
Matlab m-files many global test problems
Test Problems for nonsmooth optimization large and small scale instances
PSARNOT a test set for evaluating methods for nonsmooth optimization
Testcases for unconstrained optimization codes collected by Buckley
Testcases for unconstrained optimization codes generators by Nielsen (Matlab)
The More-Garbow-Hillstrom collection providing first and second derivatives
The More-Garbow-Hillstrom collection in Matlab
The GKLS generator includes RNG, docs (C)
Global trajectory optimization problems Matlab/C++/Python

There are more codes for unconstrained problems in the general collections listed below.

Testcases for nonlinear systems of equations and nonlinear least squares problems

Netlib/Minpack dense problems
Coprin bench page small polynomial and nonpolynomial systems
Luksan's test problems sparse problems; also others
polynomial systems with results for PHCpack

Testcases for QP

convex QP testproblems in extended MPS format or here or in AMPL format
JCastro Some of these problems are QPs in MPS or AMPL NL format
QPlib (multiple formats) Both convex, non convex, continuous, discrete instances
Benchmark examples for the OSQP solver some benchmark instances and scripts to compare QP solvers

Testcases for semidefinite and second-order cone programming

CBLIB Conic Benchmark Library with new format and conversion scripts (incl discrete problems)
SDPLIB SDP test problems in sparse SDPA-format
SQL SQL problems (DIMACS Challenge), SeDuMi and DIMACS graph formats, in sparse SDPA format
SOTIROV A new library of structured semidefinite programming instances (sparse SDPA format)
RDM Semidefinite programming instances from quantum chemistry (sparse SDPA format)
KOCVARA sparse SDPs from structural optimization (in sparse SDPA and Matlab binary format)
QAP SDP relaxations of QAP problems by Rendl, Sotirov, and Wolkowicz (in sparse SDPA and Matlab binary format)
SOCP Second-order cone problems from 7th DIMACS Challenge (in extended MPS format)
SDP More SDP problems (in sparse SDPA and SeDuMi format)
COMPlib a collection of test examples for nonlinear semidefinite programs, control system design and related problems

Testcases for general nonlinear Programming

The following collection is written in standard f77 with milstd1753 extensions. It uses a problem formulation for nonlinear programming f(x)=min subject to h(x)=0 and g(x)>=0, where h and g are general smooth vector functions. there is also an interface for the format used e.g. by codes like NPSOL, MINOS and SNOPT. The collection contains all examples of the two collections assembled by Schittkowski resp. Hock and Schittkowski, most of Himmelblau and Dembo and some additional ones. The code DONLP2 solves them all but (purposely) one successfully.

testcases for sparse and nonsmooth optimization, including nonlinear systems of equations

Testcases for PDE constrained optimization

Elliptic and parabolic cases coded in AMPL
OPTPDE Mathematical description and references

Testcases for parameter estimation

1300 test problems from data fitting, system identification, nonlinear regression

Testcases for various discrete optimization problems

Biq Mac Library Binary Quadratic and Max Cut Library
Multicommodity problems problems, generators, format converter (C++)
Capacited Linear Multicommodity Flow Problems problems, generators, format converter (C++)
frequency assignment problems benchmarks, other info on FAP
traffic assignment problems info on previously used testproblems
TSPLIB library of traveling salesman, Hamiltonian cycle, sequential ordering, and capacitated vehicle routing problems
LOLIB library of linear ordering problems
UNIBO extensive library of codes and instances
VRPTW VRPTW Benchmark Problems
QAPLIB QAP Library and related links
SATLIB SAT benchmarks, solvers, links etc.
SATlive SAT benchmarks, solvers, links etc., updated
SAT Competition Web Page annual evaluation with many examples
BHOSLIB Benchmarks with hidden optimum solutions for graph problems
CSPLIB a problem library for constraints
SteinLib a collection of Steiner tree problems in graphs
PSPLIB Project Scheduling Library
Taillard's instances QAP, Scheduling, VRP
OR-Library testdata for a variety of OR problems
0-1 Constraint Satisfaction Benchmarks realistic cases in various formats

The error free formulation of large probems by direct coding in some programming language is fatiguing. Special coding devices are of great help here. The SIF (=standard input format) developed by Conn, Gould and Toint is one of them. The following collection contains nearly a thousand problems (with the additional possibility to vary dimension) coded in SIF. The selection tool allows you to extract subcollections of specific properties.

Testcases coded in special format

CUTEst Constrained and Unconstrained Testing Environment
including large scale testcases, in SIF format
Libopt environment for testing solvers on heterogeneous collections of problems

AMPL and GAMS are modelling languages which allow a user to formulate problems in terms very near the original problem and transform this into a format required by specific solvers via specialized interfaces of which no knowledge is required by the user. They also provide automatically analytic derivatives.

Our AMPL collection files for various optimization problems
COCONUT Benchmark NLPs and constraint satisfaction problems
AMPL models and interfaces the collection at netlib
AMPL model files R. Vanderbei's collection
COPS Large-Scale Nonlinearly Constrained Optimization Problems (22 documented examples as AMPL models, 17 also in GAMS)
NLP and MINLP test problems BARON, GAMS, and Matlab format
polynomial MIP problems PIP, ZIMPL, GAMS, AMPL formats
MINLPLIB various formats, includes other libraries
MINLP testproblems AMPL format
MP's with equilibrium constraints AMPL format
Equilibrium problems with equilibrium constraints AMPL format
Multiobjective Optimization Testproblems AMPL format