Different fit indices take into account different aspects of the model, and have different pros and cons. I believe it largely comes down to your rationale for which indices you base your fit on, and which you report. Though there are some that are more commonly reported/expected than others.

Attached is a handy cheatsheet from Cornell that might help you make your call.

Well, yes some references say that there must be minimum one fit to be present to say that the model is fit. Hair et al. (1995, 2010) and Holmes-Smith (2006) suggests that at least one index from each model fitness category should be used. The three model fit categories are Absolute Fit (RMSEA, GFI), Incremental Fit (AGFI, CFI, TLI, NFI) and Parsimonious Fit (Relative chi square).

RMSEA depict the fitness of the overall model. However, other fit indices like CFI, GFI, NFI, SRMR and so on must be met with their cut-offs. Check Kline, 2005

Thank you Josh Kaplan for your reference