Several clients have asked us whether the government shutdown means that USCIS is shut down as well. Short answer: NO. And that's thanks to you, the immigration applicant, actually.
Remember that large filing fee that you paid to USCIS when you filed your application? That's where USCIS gets its money to operate -- not from Congressional appropriations. So USCIS will keep operating almost entirely normally. Your paperwork will still be processed (sometimes more slowly than ideal), and your interview will still take place. USCIS asylum offices are also open for business, including holding regularly scheduled interviews.
Immigration Court, however, will likely be shut down until the government reopens -- except for cases involving detained non-citizens. But be sure to check with your local Immigration Court before you assume that a hearing will be postponed.
As of now, U.S. consulates in foreign countries, as well as the National Visa Center, also appear to be operating normally.