What are the most widely used circuit simulators in industry?
I'm going to sorta disagree with Olin.  

If you're using a simulator for something you can do with a calculator and a piece of paper in a few minutes then you're using the simulator for the wrong purpose.  If you assume that your 'napkin math' analysis will hold up in reality you're likely working on extremely simple, basic circuits in the first place.  

More importantly you're really saying that it's OK to skip what is really the most important stage of pre-prototype design verification.  This is a really bad idea if you're working on anything even moderately complex and very much can come back to bite you even on simple circuits.  I've seen even the simplest IR transmitter oscillate due to parasitics.

Additionally, a huge use case of simulation that is a real pain to do with just a calculator is Montecarlo analysis.  Almost every simulator supports this and it is very important for production designs.

It's actually very rare that a simulator will not give you more insight into a real circuit than a 2-minute, mostly intuition-based, analysis of the circuit will.  A couple of hours generating the simulation can easily save you days waiting to get a prototype back only to find out that through some awkward component or parasitic interactions your theoretically perfect transmitter is just a lousy oscillator.

As an example, taken from an Analog Devices app note:



On the left we see a basic op amp circuit.  On the right we see what this circuit would look like if one considers basic PCB parasitic effects.  

No question that with 60 seconds and a calculator you can figure out what the circuit on the left is doing.  

However, that is no substitute for producing a more complex model of the real circuit in the real application such as the circuit to the right.  The right hand circuit is far from easy to analyze manually without hand-waving away components as irrelevant.

Additionally a proper simulation is going to use more realistic models for components, rather than the ideal models, which for a circuit of any complexity or speed is critically important to understand and analyze.

As to the original question:

Most circuit simulators are at least related to SPICE and many share a compatible or close to compatible model format.  Additionally there are many other simulators which specialize in particular fields.  Notably RF/microwave simulation, digital logic simulation, etc.

The most common simulators I've run into:


PSPICE - part of Cadence's OrCAD design package
Spectre - Mixed signal and RF simulator from Cadence (maybe the most common)
NI MutiSim - National Instruments simulation package
HSPICE - commerical SPICE implementation by Synopsis, also very popular
XSPICE - extended version of SPICE3, Altium uses this
SIMetrix - SPICE derived analog simulation


Which one(s) you will see in a given company is usually a function of their field of specialty (analog, mixed-signal, RF, etc), what integrates well in their chosen development environment and what they are historically comfortable with.