How can I list the available Cipher algorithms?

To get a list of available cipher transformation names, use this code snippet:

Set<String> algs = new TreeSet<>();
for (Provider provider : Security.getProviders()) {
    provider.getServices().stream()
                          .filter(s -> "Cipher".equals(s.getType()))
                          .map(Service::getAlgorithm)
                          .forEach(algs::add);
}
algs.forEach(System.out::println);

Those names are compatible to be called via Cipher.getInstance(). If an entry is missing the feedback mode or padding scheme, the JVM will fallback to the corresponding default.

AES
AESWrap
AESWrap_128
AESWrap_192
AESWrap_256
AES_128/CBC/NoPadding
AES_128/CFB/NoPadding
AES_128/ECB/NoPadding
AES_128/GCM/NoPadding
AES_128/OFB/NoPadding
AES_192/CBC/NoPadding
AES_192/CFB/NoPadding
AES_192/ECB/NoPadding
AES_192/GCM/NoPadding
AES_192/OFB/NoPadding
AES_256/CBC/NoPadding
AES_256/CFB/NoPadding
AES_256/ECB/NoPadding
AES_256/GCM/NoPadding
AES_256/OFB/NoPadding
ARCFOUR
Blowfish
ChaCha20
ChaCha20-Poly1305
DES
DESede
DESedeWrap
PBEWithHmacSHA1AndAES_128
PBEWithHmacSHA1AndAES_256
PBEWithHmacSHA224AndAES_128
PBEWithHmacSHA224AndAES_256
PBEWithHmacSHA256AndAES_128
PBEWithHmacSHA256AndAES_256
PBEWithHmacSHA384AndAES_128
PBEWithHmacSHA384AndAES_256
PBEWithHmacSHA512AndAES_128
PBEWithHmacSHA512AndAES_256
PBEWithMD5AndDES
PBEWithMD5AndTripleDES
PBEWithSHA1AndDESede
PBEWithSHA1AndRC2_128
PBEWithSHA1AndRC2_40
PBEWithSHA1AndRC4_128
PBEWithSHA1AndRC4_40
RC2
RSA
RSA/ECB/PKCS1Padding

see also JDK doc cipher algorithm names

To get a list of available TLS cipher suites, use this one-liner:

$ jrunscript -e "java.util.Arrays.asList(javax.net.ssl.SSLServerSocketFactory.getDefault().getSupportedCipherSuites()).stream().forEach(println)"

Which outputs for JDK11:

TLS_AES_128_GCM_SHA256
TLS_AES_256_GCM_SHA384
TLS_ECDHE_ECDSA_WITH_AES_256_GCM_SHA384
TLS_ECDHE_ECDSA_WITH_AES_128_GCM_SHA256
TLS_ECDHE_RSA_WITH_AES_256_GCM_SHA384
TLS_RSA_WITH_AES_256_GCM_SHA384
TLS_ECDH_ECDSA_WITH_AES_256_GCM_SHA384
TLS_ECDH_RSA_WITH_AES_256_GCM_SHA384
TLS_DHE_RSA_WITH_AES_256_GCM_SHA384
TLS_DHE_DSS_WITH_AES_256_GCM_SHA384
TLS_ECDHE_RSA_WITH_AES_128_GCM_SHA256
TLS_RSA_WITH_AES_128_GCM_SHA256
TLS_ECDH_ECDSA_WITH_AES_128_GCM_SHA256
TLS_ECDH_RSA_WITH_AES_128_GCM_SHA256
TLS_DHE_RSA_WITH_AES_128_GCM_SHA256
TLS_DHE_DSS_WITH_AES_128_GCM_SHA256
TLS_ECDHE_ECDSA_WITH_AES_256_CBC_SHA384
TLS_ECDHE_RSA_WITH_AES_256_CBC_SHA384
TLS_RSA_WITH_AES_256_CBC_SHA256
TLS_ECDH_ECDSA_WITH_AES_256_CBC_SHA384
TLS_ECDH_RSA_WITH_AES_256_CBC_SHA384
TLS_DHE_RSA_WITH_AES_256_CBC_SHA256
TLS_DHE_DSS_WITH_AES_256_CBC_SHA256
TLS_ECDHE_ECDSA_WITH_AES_256_CBC_SHA
TLS_ECDHE_RSA_WITH_AES_256_CBC_SHA
TLS_RSA_WITH_AES_256_CBC_SHA
TLS_ECDH_ECDSA_WITH_AES_256_CBC_SHA
TLS_ECDH_RSA_WITH_AES_256_CBC_SHA
TLS_DHE_RSA_WITH_AES_256_CBC_SHA
TLS_DHE_DSS_WITH_AES_256_CBC_SHA
TLS_ECDHE_ECDSA_WITH_AES_128_CBC_SHA256
TLS_ECDHE_RSA_WITH_AES_128_CBC_SHA256
TLS_RSA_WITH_AES_128_CBC_SHA256
TLS_ECDH_ECDSA_WITH_AES_128_CBC_SHA256
TLS_ECDH_RSA_WITH_AES_128_CBC_SHA256
TLS_DHE_RSA_WITH_AES_128_CBC_SHA256
TLS_DHE_DSS_WITH_AES_128_CBC_SHA256
TLS_ECDHE_ECDSA_WITH_AES_128_CBC_SHA
TLS_ECDHE_RSA_WITH_AES_128_CBC_SHA
TLS_RSA_WITH_AES_128_CBC_SHA
TLS_ECDH_ECDSA_WITH_AES_128_CBC_SHA
TLS_ECDH_RSA_WITH_AES_128_CBC_SHA
TLS_DHE_RSA_WITH_AES_128_CBC_SHA
TLS_DHE_DSS_WITH_AES_128_CBC_SHA

Once I have a list of providers, as described in JB Nizet's post, I still don't have a list of algorithms. I found that each Provider functions as a Properties object, and the Properties encode the algorithm names. I'm not entirely clear on if this is the correct way to look for them or not, and what exactly all the other properties mean, but I just wrote a routine that spewed all properties to System.out and grepped for various strings describing what I was looking for until I found it.

import java.security.*;

for (Provider provider: Security.getProviders()) {
  System.out.println(provider.getName());
  for (String key: provider.stringPropertyNames())
    System.out.println("\t" + key + "\t" + provider.getProperty(key));
}

The doc of Cipher.getInstance() says:

Note that the list of registered providers may be retrieved via the Security.getProviders() method

Clicking on the link leads to the doc of Provider, which has a method getServices() documented by :

Get an unmodifiable Set of all services supported by this Provider.

And clicking on the link leads to the doc of Provider which has a getAlgorithm() method.

Note that this is a very empirical method. A more logical method would be to read the documentation of the crypto libraries you're using. It must contain the list of supported algorithms.

Whatever the method you choose, reading the documentation helps a lot.