There are several synonyms for the phrase "show a smile". One could say "flash a grin" to imply a bigger, toothy smile. Alternatively, someone might suggest "beam" or "radiate" a smile to convey a sense of warmth and positivity. A more casual phrase might be "smirk" or "grin broadly" to suggest a playful or mischievous smile. For a more reserved or polite smile, one might say "smile demurely" or "offer a polite smile". There are countless variations on the phrase "show a smile" that can tweak the meaning or tone of the emotion being conveyed, making it an incredibly versatile expression in English.