Scripted Jenkins pipeline: continue on fail

The usual approach is to wrap your steps within a try block.

try {
  sh "..."
} catch (err) {
  echo "something failed"
}
// cleanup
sh "rm -rf *"

To ease the pain and make the pipeline code more readable, I've encapsulated this in another method here in my global library code.

Another approach, esp. created because of this very issue, are the declarative pipelines (blog, presentation).

The accepted answer wouldn't fail the stage or even mark it as unstable. It is now possible to fail a stage, continue the execution of the pipeline and choose the result of the build:

pipeline {
    agent any
    stages {
        stage('1') {
            steps {
                sh 'exit 0'
            }
        }
        stage('2') {
            steps {
                catchError(buildResult: 'SUCCESS', stageResult: 'FAILURE') {
                    sh "exit 1"
                }
            }
        }
        stage('3') {
            steps {
                sh 'exit 0'
            }
        }
    }
}

In the example above, all stages will execute, the pipeline will be successful, but stage 2 will show as failed:

As you might have guessed, you can freely choose the buildResult and stageResult, in case you want it to be unstable or anything else. You can even fail the build and continue the execution of the pipeline.

Just make sure your Jenkins is up to date, since this is a fairly new feature.